Theorem ifpdfan2 37807

Description: Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)

Assertion

Ref	Expression
ifpdfan2	|- ( ( ph /\ ps ) <-> if- ( ph , ps , ph ) )

Proof of Theorem ifpdfan2

Step	Hyp	Ref	Expression
1		id 22	. . . 4 |- ( ph -> ph )
2	1	notnoti 137	. . 3 |- -. -. ( ph -> ph )
3	2	biorfi 422	. 2 |- ( ( ph /\ ps ) <-> ( ( ph /\ ps ) \/ -. ( ph -> ph ) ) )
4		dfifp6 1018	. 2 |- (if- ( ph , ps , ph ) <-> ( ( ph /\ ps ) \/ -. ( ph -> ph ) ) )
5	3, 4	bitr4i 267	1 |- ( ( ph /\ ps ) <-> if- ( ph , 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:  ifpancor  37808