Theorem ifpim4 37843

Description: Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)

Assertion

Ref	Expression
ifpim4	|- ( ( ph -> ps ) <-> if- ( ps , ps , -. ph ) )

Proof of Theorem ifpim4

Step	Hyp	Ref	Expression
1		simpr 477	. 2 |- ( ( ph /\ ps ) -> ps )
2		olc 399	. 2 |- ( ps -> ( ph \/ ps ) )
3		ifpim23g 37840	. 2 |- ( ( ( ph -> ps ) <-> if- ( ps , ps , -. ph ) ) <-> ( ( ( ph /\ ps ) -> ps ) /\ ( ps -> ( ph \/ ps ) ) ) )
4	1, 2, 3	mpbir2an 955	1 |- ( ( ph -> ps ) <-> if- ( ps , ps , -. 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:  ifpnim2  37844