Nuprl Lemma : xxsym_functionality_wrt_breqv
∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ]. ((R <≡>{T} R') ⇒ (sym(T;R) ⇐⇒ sym(T;R')))
Proof
Definitions occuring in Statement :
xxsym: sym(T;E),
binrel_eqv: E <≡>{T} E',
uall: ∀[x:A]. B[x],
prop: ℙ,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
xxsym: sym(T;E),
sym: Sym(T;x,y.E[x; y]),
all: ∀x:A. B[x],
member: t ∈ T,
binrel_eqv: E <≡>{T} E',
prop: ℙ,
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
xxsym_wf,
binrel_eqv_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
applyEquality,
because_Cache,
lemma_by_obid,
isectElimination,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R,R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. ((R <\mequiv{}>\{T\} R') {}\mRightarrow{} (sym(T;R) \mLeftarrow{}{}\mRightarrow{} sym(T;R')))
Date html generated:
2016_05_15-PM-00_00_51
Last ObjectModification:
2015_12_26-PM-11_26_48
Theory : gen_algebra_1
Home
Index