Nuprl Lemma : ifthenelse_functionality_wrt_iff3

b1,b2:𝔹.  ∀[p,q1,q2:ℙ].  (b1 b2  {q1 ⇐⇒ q2}  {if b1 then q1 else fi  ⇐⇒ if b2 then q2 else fi })


Proof




Definitions occuring in Statement :  ifthenelse: if then else fi  bool: 𝔹 uall: [x:A]. B[x] prop: guard: {T} all: x:A. B[x] iff: ⇐⇒ Q implies:  Q equal: t ∈ T
Definitions unfolded in proof :  guard: {T} all: x:A. B[x] uall: [x:A]. B[x] implies:  Q iff: ⇐⇒ Q and: P ∧ Q member: t ∈ T bool: 𝔹 unit: Unit it: btrue: tt uiff: uiff(P;Q) uimplies: supposing a ifthenelse: if then else fi  assert: b true: True prop: rev_implies:  Q sq_type: SQType(T) bfalse: ff exists: x:A. B[x] or: P ∨ Q bnot: ¬bb false: False
Lemmas referenced :  eqtt_to_assert subtype_base_sq bool_subtype_base iff_imp_equal_bool btrue_wf assert_wf true_wf eqff_to_assert equal_wf bool_wf bool_cases_sqequal assert_of_bnot ifthenelse_wf iff_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep lambdaFormation isect_memberFormation sqequalHypSubstitution productElimination thin independent_pairFormation cut hypothesisEquality because_Cache unionElimination equalityElimination introduction extract_by_obid isectElimination hypothesis independent_isectElimination independent_functionElimination instantiate natural_numberEquality dependent_functionElimination equalityTransitivity equalitySymmetry dependent_pairFormation promote_hyp voidElimination cumulativity universeEquality

Latex:
\mforall{}b1,b2:\mBbbB{}.
    \mforall{}[p,q1,q2:\mBbbP{}].    (b1  =  b2  {}\mRightarrow{}  \{q1  \mLeftarrow{}{}\mRightarrow{}  q2\}  {}\mRightarrow{}  \{if  b1  then  q1  else  p  fi    \mLeftarrow{}{}\mRightarrow{}  if  b2  then  q2  else  p  fi  \})



Date html generated: 2017_04_14-AM-07_30_08
Last ObjectModification: 2017_02_27-PM-02_58_56

Theory : bool_1


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