Theorem ifpfal 996

Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 3610. This is essentially dedlemb 976. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 25-Jun-2020.)

Assertion

Ref	Expression
ifpfal	|- ( -. ph -> (if- ( ph , ps , ch ) <-> ch ) )

Proof of Theorem ifpfal

Step	Hyp	Ref	Expression
1		df-ifp 984	. . 3 |- (if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
2		ancom 266	. . . 4 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
3		ancom 266	. . . 4 |- ( ( -. ph /\ ch ) <-> ( ch /\ -. ph ) )
4	2, 3	orbi12i 769	. . 3 |- ( ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) )
5	1, 4	bitri 184	. 2 |- (if- ( ph , ps , ch ) <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) )
6		dedlemb 976	. 2 |- ( -. ph -> ( ch <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) )
7	5, 6	bitr4id 199	1 |- ( -. ph -> (if- ( ph , ps , ch ) <-> ch ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  if-wif 983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-ifp 984

This theorem is referenced by:  ifpiddc  997