Theorem dfifp6 1018

Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)

Assertion

Ref	Expression
dfifp6	|- (if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ -. ( ch -> ph ) ) )

Proof of Theorem dfifp6

Step	Hyp	Ref	Expression
1		df-ifp 1013	. 2 |- (if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
2		ancom 466	. . . 4 |- ( ( -. ph /\ ch ) <-> ( ch /\ -. ph ) )
3		annim 441	. . . 4 |- ( ( ch /\ -. ph ) <-> -. ( ch -> ph ) )
4	2, 3	bitri 264	. . 3 |- ( ( -. ph /\ ch ) <-> -. ( ch -> ph ) )
5	4	orbi2i 541	. 2 |- ( ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) <-> ( ( ph /\ ps ) \/ -. ( ch -> ph ) ) )
6	1, 5	bitri 264	1 |- (if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ -. ( ch -> ph ) ) )

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:  dfifp7  1019  ifpdfan2  37807