Theorem dfifp3 1427

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

Assertion

Ref	Expression
dfifp3	|- (if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( ph \/ ch ) ) )

Proof of Theorem dfifp3

Step	Hyp	Ref	Expression
1		dfifp2 1426	. 2 |- (if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )
2		pm4.64 374	. . 3 |- ( ( -. ph -> ch ) <-> ( ph \/ ch ) )
3	2	anbi2i 699	. 2 |- ( ( ( ph -> ps ) /\ ( -. ph -> ch ) ) <-> ( ( ph -> ps ) /\ ( ph \/ ch ) ) )
4	1, 3	bitri 253	1 |- (if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( ph \/ ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371  if-wif 1424

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 189  df-or 372  df-an 373  df-ifp 1425

This theorem is referenced by:  dfifp4  1428  ifptru  1435