Theorem ifpnot23 37823

Description: Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)

Assertion

Ref	Expression
ifpnot23	|- ( -. if- ( ph , ps , ch ) <-> if- ( ph , -. ps , -. ch ) )

Proof of Theorem ifpnot23

Step	Hyp	Ref	Expression
1		ianor 509	. . . 4 |- ( -. ( ph /\ ps ) <-> ( -. ph \/ -. ps ) )
2		pm4.55 515	. . . 4 |- ( -. ( -. ph /\ ch ) <-> ( ph \/ -. ch ) )
3	1, 2	anbi12i 733	. . 3 |- ( ( -. ( ph /\ ps ) /\ -. ( -. ph /\ ch ) ) <-> ( ( -. ph \/ -. ps ) /\ ( ph \/ -. ch ) ) )
4		ioran 511	. . 3 |- ( -. ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) <-> ( -. ( ph /\ ps ) /\ -. ( -. ph /\ ch ) ) )
5		dfifp4 1016	. . 3 |- (if- ( ph , -. ps , -. ch ) <-> ( ( -. ph \/ -. ps ) /\ ( ph \/ -. ch ) ) )
6	3, 4, 5	3bitr4i 292	. 2 |- ( -. ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) <-> if- ( ph , -. ps , -. ch ) )
7		df-ifp 1013	. 2 |- (if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
8	6, 7	xchnxbir 323	1 |- ( -. if- ( ph , ps , ch ) <-> if- ( ph , -. ps , -. ch ) )

Syntax hints:   -. wn 3   <-> 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:  ifpnotnotb  37824  ifpnorcor  37825  ifpnancor  37826  ifpnot23b  37827  ifpnot23c  37829  ifpnot23d  37830  ifpdfnan  37831  ifpdfxor  37832  ifpor123g  37853