Theorem ifpbi123d 998

Description: Equivalence deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 17-Apr-2024.)

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 473	. . 3 |- ( ph -> ( ( ps /\ ch ) <-> ( ta /\ et ) ) )
4	1	notbid 671	. . . 4 |- ( ph -> ( -. ps <-> -. ta ) )
5		ifpbi123d.3	. . . 4 |- ( ph -> ( th <-> ze ) )
6	4, 5	anbi12d 473	. . 3 |- ( ph -> ( ( -. ps /\ th ) <-> ( -. ta /\ ze ) ) )
7	3, 6	orbi12d 798	. 2 |- ( ph -> ( ( ( ps /\ ch ) \/ ( -. ps /\ th ) ) <-> ( ( ta /\ et ) \/ ( -. ta /\ ze ) ) ) )
8		df-ifp 984	. 2 |- (if- ( ps , ch , th ) <-> ( ( ps /\ ch ) \/ ( -. ps /\ th ) ) )
9		df-ifp 984	. 2 |- (if- ( ta , et , ze ) <-> ( ( ta /\ et ) \/ ( -. ta /\ ze ) ) )
10	7, 8, 9	3bitr4g 223	1 |- ( ph -> (if- ( ps , ch , th ) <-> if- ( ta , et , ze ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  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-ifp 984

This theorem is referenced by:  ifpbi23d  999