Nuprl Lemma : face-term-and-implies2
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. Gamma ⊢ ((phi ∧ psi) ⇒ psi)
Proof
Definitions occuring in Statement :
face-term-implies: Gamma ⊢ (phi ⇒ psi),
face-and: (a ∧ b),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
all: ∀x:A. B[x]
Lemmas referenced :
face-term-implies_wf,
squash_wf,
true_wf,
cubical-term_wf,
face-type_wf,
cubical_set_wf,
face-and-com,
subtype_rel_self,
iff_weakening_equal,
face-term-and-implies1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
instantiate,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
inhabitedIsType,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. Gamma \mvdash{} ((phi \mwedge{} psi) {}\mRightarrow{} psi)
Date html generated:
2020_05_20-PM-02_47_16
Last ObjectModification:
2020_04_04-PM-05_01_22
Theory : cubical!type!theory
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