Theorem ifpnst 994

Description: Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.) (Proof shortened by Wolf Lammen, 5-May-2024.)

Assertion

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

Proof of Theorem ifpnst

Step	Hyp	Ref	Expression
1		ifpdc 985	. . 3 |- (if- ( ph , ps , ch ) -> DECID ph )
2	1	adantl 277	. 2 |- ( (STAB ph /\ if- ( ph , ps , ch ) ) -> DECID ph )
3		ifpdc 985	. . 3 |- (if- ( -. ph , ch , ps ) -> DECID -. ph )
4		stdcndc 850	. . . 4 |- ( (STAB ph /\ DECID -. ph ) <-> DECID ph )
5	4	biimpi 120	. . 3 |- ( (STAB ph /\ DECID -. ph ) -> DECID ph )
6	3, 5	sylan2 286	. 2 |- ( (STAB ph /\ if- ( -. ph , ch , ps ) ) -> DECID ph )
7		dfifp5dc 990	. . . 4 |- (DECID ph -> (if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( -. ph -> ch ) ) ) )
8	7	biancomd 271	. . 3 |- (DECID ph -> (if- ( ph , ps , ch ) <-> ( ( -. ph -> ch ) /\ ( -. ph \/ ps ) ) ) )
9		dcn 847	. . . 4 |- (DECID ph -> DECID -. ph )
10		dfifp3dc 988	. . . 4 |- (DECID -. ph -> (if- ( -. ph , ch , ps ) <-> ( ( -. ph -> ch ) /\ ( -. ph \/ ps ) ) ) )
11	9, 10	syl 14	. . 3 |- (DECID ph -> (if- ( -. ph , ch , ps ) <-> ( ( -. ph -> ch ) /\ ( -. ph \/ ps ) ) ) )
12	8, 11	bitr4d 191	. 2 |- (DECID ph -> (if- ( ph , ps , ch ) <-> if- ( -. ph , ch , ps ) ) )
13	2, 6, 12	pm5.21nd 921	1 |- (STAB ph -> (if- ( ph , ps , ch ) <-> if- ( -. ph , ch , ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  STAB wstab 835  DECID wdc 839  if-wif 983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-ifp 984

This theorem is referenced by: (None)