Theorem ifpnot23d 37830

Description: Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)

Assertion

Ref	Expression
ifpnot23d	|- ( -. if- ( ph , -. ps , -. ch ) <-> if- ( ph , ps , ch ) )

Proof of Theorem ifpnot23d

Step	Hyp	Ref	Expression
1		ifpnot23 37823	. 2 |- ( -. if- ( ph , -. ps , -. ch ) <-> if- ( ph , -. -. ps , -. -. ch ) )
2		notnotb 304	. . 3 |- ( ps <-> -. -. ps )
3		notnotb 304	. . 3 |- ( ch <-> -. -. ch )
4		ifpbi23 37817	. . 3 |- ( ( ( ps <-> -. -. ps ) /\ ( ch <-> -. -. ch ) ) -> (if- ( ph , ps , ch ) <-> if- ( ph , -. -. ps , -. -. ch ) ) )
5	2, 3, 4	mp2an 708	. 2 |- (if- ( ph , ps , ch ) <-> if- ( ph , -. -. ps , -. -. ch ) )
6	1, 5	bitr4i 267	1 |- ( -. if- ( ph , -. ps , -. ch ) <-> if- ( ph , ps , ch ) )

Syntax hints:   -. wn 3   <-> wb 196  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:  ifpororb  37850