Theorem ifpbi123d 1027

Description: Equality deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.)

Hypotheses

Ref	Expression
ifpbi123d.1	|- ( ph -> ( ps <-> ta ) )
ifpbi123d.2	|- ( ph -> ( ch <-> et ) )
ifpbi123d.3	|- ( ph -> ( th <-> ze ) )

Assertion

Ref	Expression
ifpbi123d	|- ( ph -> (if- ( ps , ch , th ) <-> if- ( ta , et , ze ) ) )

Proof of Theorem ifpbi123d

Step	Hyp	Ref	Expression
1		ifpbi123d.1	. . . 4 |- ( ph -> ( ps <-> ta ) )
2		ifpbi123d.2	. . . 4 |- ( ph -> ( ch <-> et ) )
3	1, 2	anbi12d 747	. . 3 |- ( ph -> ( ( ps /\ ch ) <-> ( ta /\ et ) ) )
4	1	notbid 308	. . . 4 |- ( ph -> ( -. ps <-> -. ta ) )
5		ifpbi123d.3	. . . 4 |- ( ph -> ( th <-> ze ) )
6	4, 5	anbi12d 747	. . 3 |- ( ph -> ( ( -. ps /\ th ) <-> ( -. ta /\ ze ) ) )
7	3, 6	orbi12d 746	. 2 |- ( ph -> ( ( ( ps /\ ch ) \/ ( -. ps /\ th ) ) <-> ( ( ta /\ et ) \/ ( -. ta /\ ze ) ) ) )
8		df-ifp 1013	. 2 |- (if- ( ps , ch , th ) <-> ( ( ps /\ ch ) \/ ( -. ps /\ th ) ) )
9		df-ifp 1013	. 2 |- (if- ( ta , et , ze ) <-> ( ( ta /\ et ) \/ ( -. ta /\ ze ) ) )
10	7, 8, 9	3bitr4g 303	1 |- ( ph -> (if- ( ps , ch , th ) <-> if- ( ta , et , ze ) ) )

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:  wkslem1  26503  wkslem2  26504  wksfval  26505  iswlk  26506  wlkres  26567  redwlk  26569  wlkp1lem8  26577  crctcshwlkn0lem4  26705  crctcshwlkn0lem5  26706  crctcshwlkn0lem6  26707  1wlkdlem4  27000