Theorem ifptru 1023

Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4092. This is essentially dedlema 1002. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)

Assertion

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

Proof of Theorem ifptru

Step	Hyp	Ref	Expression
1		biimt 350	. 2 |- ( ph -> ( ps <-> ( ph -> ps ) ) )
2		orc 400	. . . 4 |- ( ph -> ( ph \/ ch ) )
3	2	biantrud 528	. . 3 |- ( ph -> ( ( ph -> ps ) <-> ( ( ph -> ps ) /\ ( ph \/ ch ) ) ) )
4		dfifp3 1015	. . 3 |- (if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( ph \/ ch ) ) )
5	3, 4	syl6bbr 278	. 2 |- ( ph -> ( ( ph -> ps ) <-> if- ( ph , ps , ch ) ) )
6	1, 5	bitr2d 269	1 |- ( ph -> (if- ( ph , ps , ch ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384  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:  ifpfal  1024  ifpid  1025  elimh  1030  dedt  1031  wlkl1loop  26534  lfgrwlkprop  26584  eupth2lem3lem3  27090