Theorem ifpfal 1024

Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4095. This is essentially dedlemb 1003. (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		ifpn 1022	. 2 |- (if- ( ph , ps , ch ) <-> if- ( -. ph , ch , ps ) )
2		ifptru 1023	. 2 |- ( -. ph -> (if- ( -. ph , ch , ps ) <-> ch ) )
3	1, 2	syl5bb 272	1 |- ( -. ph -> (if- ( ph , ps , ch ) <-> ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196  if-wif 1012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by:  ifpid  1025  elimh  1030  wlkdlem4  26582  lfgriswlk  26585  2pthnloop  26627  eupth2lem3lem4  27091