Nuprl Lemma : connex_functionality_wrt_implies
∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ]. ((∀x,y:T. {R[x;y] ⇒ R'[x;y]}) ⇒ {Connex(T;x,y.R[x;y]) ⇒ Connex(T;x,y.R'[x;y])})
Proof
Definitions occuring in Statement :
connex: Connex(T;x,y.R[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
connex: Connex(T;x,y.R[x; y]),
guard: {T},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
uimplies: b supposing a
Lemmas referenced :
all_wf,
or_wf,
all_functionality_wrt_implies,
or_functionality_wrt_implies
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
functionEquality,
universeEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R,R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:T. \{R[x;y] {}\mRightarrow{} R'[x;y]\}) {}\mRightarrow{} \{Connex(T;x,y.R[x;y]) {}\mRightarrow{} Connex(T;x,y.R'[x;y])\})
Date html generated:
2016_10_21-AM-09_42_22
Last ObjectModification:
2016_08_01-PM-09_49_16
Theory : rel_1
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