Theorem dfifp7 1019

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

Assertion

Ref	Expression
dfifp7	|- (if- ( ph , ps , ch ) <-> ( ( ch -> ph ) -> ( ph /\ ps ) ) )

Proof of Theorem dfifp7

Step	Hyp	Ref	Expression
1		orcom 402	. 2 |- ( ( ( ph /\ ps ) \/ -. ( ch -> ph ) ) <-> ( -. ( ch -> ph ) \/ ( ph /\ ps ) ) )
2		dfifp6 1018	. 2 |- (if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ -. ( ch -> ph ) ) )
3		imor 428	. 2 |- ( ( ( ch -> ph ) -> ( ph /\ ps ) ) <-> ( -. ( ch -> ph ) \/ ( ph /\ ps ) ) )
4	1, 2, 3	3bitr4i 292	1 |- (if- ( ph , ps , ch ) <-> ( ( ch -> ph ) -> ( ph /\ ps ) ) )

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: (None)