Theorem ifpn 1022

Description: Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.)

Assertion

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

Proof of Theorem ifpn

Step	Hyp	Ref	Expression
1		notnotb 304	. . . 4 |- ( ph <-> -. -. ph )
2	1	imbi1i 339	. . 3 |- ( ( ph -> ps ) <-> ( -. -. ph -> ps ) )
3	2	anbi2ci 732	. 2 |- ( ( ( ph -> ps ) /\ ( -. ph -> ch ) ) <-> ( ( -. ph -> ch ) /\ ( -. -. ph -> ps ) ) )
4		dfifp2 1014	. 2 |- (if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )
5		dfifp2 1014	. 2 |- (if- ( -. ph , ch , ps ) <-> ( ( -. ph -> ch ) /\ ( -. -. ph -> ps ) ) )
6	3, 4, 5	3bitr4i 292	1 |- (if- ( ph , ps , ch ) <-> if- ( -. ph , ch , ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ 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:  ifpfal  1024  ifpdfbi  37818  ifpxorcor  37821