Nuprl Lemma : test-exists-example1
∀[D:ℙ]. ∀[P,Q,R:D ⟶ ℙ].
((∀x:D. ((R x) ⇒ (Q x))) ⇒ (∀x:D. ((P x) ⇒ ((Q x) ∨ (R x)))) ⇒ (∃x:D. (P x)) ⇒ (∃x:D. (Q x)))
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
or: P ∨ Q
Lemmas referenced :
exists_wf,
all_wf,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
hypothesisEquality,
because_Cache,
applyEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
hypothesis,
functionEquality,
isect_memberFormation,
lambdaFormation,
rename,
productElimination,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
cumulativity,
universeEquality
Latex:
\mforall{}[D:\mBbbP{}]. \mforall{}[P,Q,R:D {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:D. ((R x) {}\mRightarrow{} (Q x))) {}\mRightarrow{} (\mforall{}x:D. ((P x) {}\mRightarrow{} ((Q x) \mvee{} (R x)))) {}\mRightarrow{} (\mexists{}x:D. (P x)) {}\mRightarrow{} (\mexists{}x:D. (Q x)))
Date html generated:
2016_05_15-PM-03_18_54
Last ObjectModification:
2015_12_27-PM-01_03_23
Theory : general
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