Theorem ifnebibdc 3648

Description: The converse of ifbi 3623 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)

Assertion

Ref	Expression
ifnebibdc	|- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( if ( ph , A , B ) = if ( ps , A , B ) <-> ( ph <-> ps ) ) )

Proof of Theorem ifnebibdc

Step	Hyp	Ref	Expression
1		eqifdc 3639	. . . 4 |- (DECID ps -> ( if ( ph , A , B ) = if ( ps , A , B ) <-> ( ( ps /\ if ( ph , A , B ) = A ) \/ ( -. ps /\ if ( ph , A , B ) = B ) ) ) )
2	1	3ad2ant2 1043	. . 3 |- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( if ( ph , A , B ) = if ( ps , A , B ) <-> ( ( ps /\ if ( ph , A , B ) = A ) \/ ( -. ps /\ if ( ph , A , B ) = B ) ) ) )
3		ifnetruedc 3646	. . . . . . . . 9 |- ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) -> ph )
4	3	3expia 1229	. . . . . . . 8 |- ( (DECID ph /\ A =/= B ) -> ( if ( ph , A , B ) = A -> ph ) )
5	4	3adant2 1040	. . . . . . 7 |- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( if ( ph , A , B ) = A -> ph ) )
6	5	adantld 278	. . . . . 6 |- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( ( ps /\ if ( ph , A , B ) = A ) -> ph ) )
7		simpl 109	. . . . . 6 |- ( ( ps /\ if ( ph , A , B ) = A ) -> ps )
8	6, 7	jca2 308	. . . . 5 |- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( ( ps /\ if ( ph , A , B ) = A ) -> ( ph /\ ps ) ) )
9		pm5.1 603	. . . . 5 |- ( ( ph /\ ps ) -> ( ph <-> ps ) )
10	8, 9	syl6 33	. . . 4 |- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( ( ps /\ if ( ph , A , B ) = A ) -> ( ph <-> ps ) ) )
11		ifnefals 3647	. . . . . . . . 9 |- ( ( A =/= B /\ if ( ph , A , B ) = B ) -> -. ph )
12	11	ex 115	. . . . . . . 8 |- ( A =/= B -> ( if ( ph , A , B ) = B -> -. ph ) )
13	12	3ad2ant3 1044	. . . . . . 7 |- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( if ( ph , A , B ) = B -> -. ph ) )
14	13	adantld 278	. . . . . 6 |- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( ( -. ps /\ if ( ph , A , B ) = B ) -> -. ph ) )
15		simpl 109	. . . . . 6 |- ( ( -. ps /\ if ( ph , A , B ) = B ) -> -. ps )
16	14, 15	jca2 308	. . . . 5 |- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( ( -. ps /\ if ( ph , A , B ) = B ) -> ( -. ph /\ -. ps ) ) )
17		pm5.21 700	. . . . 5 |- ( ( -. ph /\ -. ps ) -> ( ph <-> ps ) )
18	16, 17	syl6 33	. . . 4 |- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( ( -. ps /\ if ( ph , A , B ) = B ) -> ( ph <-> ps ) ) )
19	10, 18	jaod 722	. . 3 |- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( ( ( ps /\ if ( ph , A , B ) = A ) \/ ( -. ps /\ if ( ph , A , B ) = B ) ) -> ( ph <-> ps ) ) )
20	2, 19	sylbid 150	. 2 |- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( if ( ph , A , B ) = if ( ps , A , B ) -> ( ph <-> ps ) ) )
21		ifbi 3623	. 2 |- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) )
22	20, 21	impbid1 142	1 |- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( if ( ph , A , B ) = if ( ps , A , B ) <-> ( ph <-> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   =/= wne 2400   ifcif 3602

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  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ne 2401  df-if 3603

This theorem is referenced by:  nninfinf  10673