Theorem ifpim2 37816

Description: Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)

Assertion

Ref	Expression
ifpim2	|- ( ( ph -> ps ) <-> if- ( ps , T. , -. ph ) )

Proof of Theorem ifpim2

Step	Hyp	Ref	Expression
1		tru 1487	. . . 4 |- T.
2	1	olci 406	. . 3 |- ( -. ps \/ T. )
3	2	biantrur 527	. 2 |- ( ( ps \/ -. ph ) <-> ( ( -. ps \/ T. ) /\ ( ps \/ -. ph ) ) )
4		imor 428	. . 3 |- ( ( ph -> ps ) <-> ( -. ph \/ ps ) )
5		orcom 402	. . 3 |- ( ( -. ph \/ ps ) <-> ( ps \/ -. ph ) )
6	4, 5	bitri 264	. 2 |- ( ( ph -> ps ) <-> ( ps \/ -. ph ) )
7		dfifp4 1016	. 2 |- (if- ( ps , T. , -. ph ) <-> ( ( -. ps \/ T. ) /\ ( ps \/ -. ph ) ) )
8	3, 6, 7	3bitr4i 292	1 |- ( ( ph -> ps ) <-> if- ( ps , T. , -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384  if-wif 1012   T. wtru 1484

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  df-tru 1486

This theorem is referenced by:  ifpdfbi  37818