### Nuprl Lemma : and_functionality_wrt_uiff2

`∀[P1,P2,Q1,Q2:ℙ].  ({uiff(P1;P2)} `` (Q1 = Q2 ∈ ℙ) `` {P1 ∧ Q1 `⇐⇒` P2 ∧ Q2})`

Proof

Definitions occuring in Statement :  uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` prop: `ℙ` guard: `{T}` iff: `P `⇐⇒` Q` implies: `P `` Q` and: `P ∧ Q` equal: `s = t ∈ T`
Definitions unfolded in proof :  guard: `{T}` uall: `∀[x:A]. B[x]` implies: `P `` Q` uiff: `uiff(P;Q)` and: `P ∧ Q` iff: `P `⇐⇒` Q` uimplies: `b supposing a` member: `t ∈ T` prop: `ℙ` rev_implies: `P `` Q`
Lemmas referenced :  equal_wf uiff_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation sqequalHypSubstitution productElimination thin independent_pairFormation independent_isectElimination hypothesis cut equalitySymmetry hyp_replacement Error :applyLambdaEquality,  hypothesisEquality productEquality cumulativity instantiate introduction extract_by_obid isectElimination universeEquality

Latex:
\mforall{}[P1,P2,Q1,Q2:\mBbbP{}].    (\{uiff(P1;P2)\}  {}\mRightarrow{}  (Q1  =  Q2)  {}\mRightarrow{}  \{P1  \mwedge{}  Q1  \mLeftarrow{}{}\mRightarrow{}  P2  \mwedge{}  Q2\})

Date html generated: 2016_10_21-AM-09_34_59
Last ObjectModification: 2016_07_12-AM-04_59_38

Theory : core_2

Home Index