### Nuprl Lemma : assert-bl-exists

`∀[T:Type]. ∀L:T List. ∀P:{x:T| (x ∈ L)}  ⟶ 𝔹.  (↑(∃x∈L.P[x])_b `⇐⇒` (∃x∈L. ↑P[x]))`

Proof

Definitions occuring in Statement :  bl-exists: `(∃x∈L.P[x])_b` l_exists: `(∃x∈L. P[x])` l_member: `(x ∈ l)` list: `T List` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` prop: `ℙ` implies: `P `` Q` bl-exists: `(∃x∈L.P[x])_b` top: `Top` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` exists: `∃x:A. B[x]` false: `False` l_member: `(x ∈ l)` cand: `A c∧ B` nat: `ℕ` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` not: `¬A` or: `P ∨ Q` uiff: `uiff(P;Q)` guard: `{T}` sq_type: `SQType(T)` btrue: `tt` true: `True`
Lemmas referenced :  list-subtype bool_subtype_base subtype_base_sq assert_elim bor_wf assert_of_bor equal_wf or_wf cons_wf cons_member nil_wf int_formula_prop_wf int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_and_lemma intformle_wf itermConstant_wf itermVar_wf intformless_wf intformand_wf satisfiable-full-omega-tt nat_properties length_of_nil_lemma false_wf bool_wf l_exists_wf l_exists_iff reduce_cons_lemma reduce_nil_lemma list_wf and_wf exists_wf l_member_wf bl-exists_wf assert_wf iff_wf list_induction
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation thin lemma_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality applyEquality setElimination rename setEquality hypothesis independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality because_Cache addLevel productElimination independent_pairFormation impliesFunctionality functionEquality universeEquality natural_numberEquality independent_isectElimination dependent_pairFormation int_eqEquality intEquality computeAll orFunctionality existsFunctionality andLevelFunctionality existsLevelFunctionality cumulativity productEquality unionElimination inlFormation inrFormation equalitySymmetry dependent_set_memberEquality equalityTransitivity levelHypothesis instantiate

Latex:
\mforall{}[T:Type].  \mforall{}L:T  List.  \mforall{}P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}.    (\muparrow{}(\mexists{}x\mmember{}L.P[x])\_b  \mLeftarrow{}{}\mRightarrow{}  (\mexists{}x\mmember{}L.  \muparrow{}P[x]))

Date html generated: 2016_05_14-PM-02_10_05
Last ObjectModification: 2016_01_15-AM-07_59_52

Theory : list_1

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