### Nuprl Lemma : all-nsub2

`∀[P:ℕ2 ⟶ ℙ]. (∀x:ℕ2. P[x] `⇐⇒` P[0] ∧ P[1])`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` less_than: `a < b` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top`
Lemmas referenced :  int_formula_prop_wf int_formula_prop_le_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformle_wf intformless_wf itermConstant_wf itermVar_wf intformeq_wf intformnot_wf intformand_wf satisfiable-full-omega-tt int_seg_properties int_subtype_base subtype_base_sq decidable__equal_int lelt_wf false_wf int_seg_wf all_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation independent_pairFormation lambdaFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality hypothesis sqequalRule lambdaEquality applyEquality hypothesisEquality productEquality dependent_set_memberEquality introduction imageMemberEquality baseClosed universeEquality because_Cache functionEquality cumulativity dependent_functionElimination setElimination rename unionElimination instantiate intEquality independent_isectElimination independent_functionElimination productElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality computeAll

Latex:
\mforall{}[P:\mBbbN{}2  {}\mrightarrow{}  \mBbbP{}].  (\mforall{}x:\mBbbN{}2.  P[x]  \mLeftarrow{}{}\mRightarrow{}  P[0]  \mwedge{}  P[1])

Date html generated: 2016_05_15-PM-03_27_26
Last ObjectModification: 2016_01_16-AM-10_48_27

Theory : general

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