Theorem ifnetruedc 3649

Description: Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)

Assertion

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

Proof of Theorem ifnetruedc

Step	Hyp	Ref	Expression
1		simp1 1023	. 2 |- ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) -> DECID ph )
2		iffalse 3613	. . . 4 |- ( -. ph -> if ( ph , A , B ) = B )
3	2	adantl 277	. . 3 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> if ( ph , A , B ) = B )
4		simpl3 1028	. . . . 5 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> if ( ph , A , B ) = A )
5		simpl2 1027	. . . . 5 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> A =/= B )
6	4, 5	eqnetrd 2426	. . . 4 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> if ( ph , A , B ) =/= B )
7	6	neneqd 2423	. . 3 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> -. if ( ph , A , B ) = B )
8	3, 7	condandc 888	. 2 |- (DECID ph -> ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) -> ph ) )
9	1, 8	mpcom 36	1 |- ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) -> ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 841   /\ w3a 1004   = wceq 1397   =/= wne 2402   ifcif 3605

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ne 2403  df-if 3606

This theorem is referenced by:  ifnebibdc  3651