Nuprl Lemma : exists-product3

[A,B,C,D:Type].  ∀P:(A × B × C × D) ⟶ ℙ'. {∃x:A × B × C × D. P[x] ⇐⇒ ∃a:A. ∃b:B. ∃c:C. ∃d:D. P[<a, b, c, d>]}


Proof




Definitions occuring in Statement :  uall: [x:A]. B[x] prop: guard: {T} so_apply: x[s] all: x:A. B[x] exists: x:A. B[x] iff: ⇐⇒ Q function: x:A ⟶ B[x] pair: <a, b> product: x:A × B[x] universe: Type
Definitions unfolded in proof :  guard: {T} uall: [x:A]. B[x] all: x:A. B[x] iff: ⇐⇒ Q and: P ∧ Q implies:  Q member: t ∈ T prop: so_lambda: λ2x.t[x] so_apply: x[s] rev_implies:  Q exists: x:A. B[x]
Lemmas referenced :  exists_wf exists-product1 iff_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation cut independent_pairFormation hypothesis thin instantiate lemma_by_obid sqequalHypSubstitution isectElimination cumulativity hypothesisEquality lambdaEquality applyEquality independent_pairEquality because_Cache addLevel productElimination independent_functionElimination productEquality dependent_functionElimination existsFunctionality impliesFunctionality levelHypothesis existsLevelFunctionality functionEquality universeEquality

Latex:
\mforall{}[A,B,C,D:Type].
    \mforall{}P:(A  \mtimes{}  B  \mtimes{}  C  \mtimes{}  D)  {}\mrightarrow{}  \mBbbP{}'.  \{\mexists{}x:A  \mtimes{}  B  \mtimes{}  C  \mtimes{}  D.  P[x]  \mLeftarrow{}{}\mRightarrow{}  \mexists{}a:A.  \mexists{}b:B.  \mexists{}c:C.  \mexists{}d:D.  P[<a,  b,  c,  d>]\}



Date html generated: 2016_05_15-PM-03_23_02
Last ObjectModification: 2015_12_27-PM-01_05_41

Theory : general


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