Nuprl Lemma : or-to-and-by-cases
∀[X:ℙ]. (Dec(X) ⇒ (∀[P,Q:ℙ]. ((P ⇒ X) ⇒ (Q ⇒ (¬X)) ⇒ {P ∨ Q ⇐⇒ (X ⇒ P) ∧ ((¬X) ⇒ Q)})))
Proof
Definitions occuring in Statement :
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
guard: {T},
decidable: Dec(P),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q
Lemmas referenced :
and_wf,
or_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
unionElimination,
thin,
hypothesis,
independent_functionElimination,
voidElimination,
hypothesisEquality,
cut,
lemma_by_obid,
isectElimination,
productElimination,
inlFormation,
inrFormation,
functionEquality,
universeEquality
Latex:
\mforall{}[X:\mBbbP{}]. (Dec(X) {}\mRightarrow{} (\mforall{}[P,Q:\mBbbP{}]. ((P {}\mRightarrow{} X) {}\mRightarrow{} (Q {}\mRightarrow{} (\mneg{}X)) {}\mRightarrow{} \{P \mvee{} Q \mLeftarrow{}{}\mRightarrow{} (X {}\mRightarrow{} P) \mwedge{} ((\mneg{}X) {}\mRightarrow{} Q)\})))
Date html generated:
2016_05_13-PM-03_17_22
Last ObjectModification:
2016_01_06-PM-05_20_15
Theory : core_2
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