Nuprl Lemma : exists_functionality_wrt_iff
∀[S,T:Type]. ∀[P,Q:S ⟶ ℙ]. (∀x:S. (P[x] ⇐⇒ Q[x])) ⇒ (∃x:S. P[x] ⇐⇒ ∃y:T. Q[y]) supposing S = T ∈ Type
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
equal_wf,
iff_wf,
all_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
lambdaFormation,
independent_pairFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
cumulativity,
hyp_replacement,
equalitySymmetry,
because_Cache,
instantiate,
universeEquality,
functionEquality,
productElimination,
dependent_pairFormation,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[S,T:Type]. \mforall{}[P,Q:S {}\mrightarrow{} \mBbbP{}]. (\mforall{}x:S. (P[x] \mLeftarrow{}{}\mRightarrow{} Q[x])) {}\mRightarrow{} (\mexists{}x:S. P[x] \mLeftarrow{}{}\mRightarrow{} \mexists{}y:T. Q[y]) supposing S = T
Date html generated:
2016_05_13-PM-03_12_28
Last ObjectModification:
2016_01_06-PM-05_24_21
Theory : core_2
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