Theorem dfifp4 1016

Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)

Assertion

Ref	Expression
dfifp4	|- (if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) )

Proof of Theorem dfifp4

Step	Hyp	Ref	Expression
1		dfifp3 1015	. 2 |- (if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( ph \/ ch ) ) )
2		imor 428	. . 3 |- ( ( ph -> ps ) <-> ( -. ph \/ ps ) )
3	2	anbi1i 731	. 2 |- ( ( ( ph -> ps ) /\ ( ph \/ ch ) ) <-> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) )
4	1, 3	bitri 264	1 |- (if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) )

Syntax hints:   -. wn 3   -> 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:  anifp  1020  ifpan123g  37803  ifpan23  37804  ifpdfor2  37805  ifpdfor  37809  ifpim1  37813  ifpnot  37814  ifpid2  37815  ifpim2  37816  ifpnot23  37823  ifpidg  37836  ifpim123g  37845  ifpimim  37854