### Nuprl Lemma : axiom-choice-quot-alt-proof

`∀T:Type`
`  (⇃(canonicalizable(T)) `` (∀X:Type. ∀P:T ⟶ X ⟶ ℙ.  ((∀f:T. ⇃(∃m:X. (P f m))) `` ⇃(∃F:T ⟶ X. ∀f:T. (P f (F f))))))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` canonicalizable: `canonicalizable(T)` prop: `ℙ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` true: `True` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  pi1: `fst(t)` guard: `{T}` choice-principle: `ChoicePrinciple(T)` uimplies: `b supposing a` so_apply: `x[s1;s2]` so_lambda: `λ2x y.t[x; y]` exists: `∃x:A. B[x]` so_apply: `x[s]` so_lambda: `λ2x.t[x]` uall: `∀[x:A]. B[x]` prop: `ℙ` rev_implies: `P `` Q` and: `P ∧ Q` iff: `P `⇐⇒` Q` member: `t ∈ T` implies: `P `` Q` all: `∀x:A. B[x]`
Lemmas referenced :  equal_wf implies-quotient-true canonicalizable_wf equiv_rel_true true_wf exists_wf quotient_wf all_wf choice-iff-canonicalizable
Rules used in proof :  equalitySymmetry equalityTransitivity rename dependent_pairFormation promote_hyp universeEquality functionEquality independent_isectElimination because_Cache functionExtensionality applyEquality lambdaEquality sqequalRule cumulativity isectElimination hypothesis independent_functionElimination productElimination hypothesisEquality thin dependent_functionElimination sqequalHypSubstitution extract_by_obid introduction cut lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}T:Type
(\00D9(canonicalizable(T))
{}\mRightarrow{}  (\mforall{}X:Type.  \mforall{}P:T  {}\mrightarrow{}  X  {}\mrightarrow{}  \mBbbP{}.    ((\mforall{}f:T.  \00D9(\mexists{}m:X.  (P  f  m)))  {}\mRightarrow{}  \00D9(\mexists{}F:T  {}\mrightarrow{}  X.  \mforall{}f:T.  (P  f  (F  f))))))

Date html generated: 2018_07_25-PM-01_50_26
Last ObjectModification: 2018_07_25-PM-00_19_24

Theory : continuity

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