Theorem ifnebibdc 3683

Description: The converse of ifbi 3658 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 3674	. . . 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 1050	. . 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 3681	. . . . . . . . 9 |- ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) -> ph )
4	3	3expia 1236	. . . . . . . 8 |- ( (DECID ph /\ A =/= B ) -> ( if ( ph , A , B ) = A -> ph ) )
5	4	3adant2 1047	. . . . . . 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		pm5.1 609	. . . . 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 3682	. . . . . . . . 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 1051	. . . . . . 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		pm5.21 707	. . . . 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 729	. . 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 3658	. 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 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   =/= wne 2420   ifcif 3635

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ne 2421  df-if 3636

This theorem is referenced by:  nninfinf  10858