Theorem ifnetruedc 3626

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 1002	. 2 |- ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) -> DECID ph )
2		iffalse 3590	. . . 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 1007	. . . . 5 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> if ( ph , A , B ) = A )
5		simpl2 1006	. . . . 5 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> A =/= B )
6	4, 5	eqnetrd 2404	. . . 4 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> if ( ph , A , B ) =/= B )
7	6	neneqd 2401	. . 3 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> -. if ( ph , A , B ) = B )
8	3, 7	condandc 885	. 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 838   /\ w3a 983   = wceq 1375   =/= wne 2380   ifcif 3582

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-11 1532  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3an 985  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-ne 2381  df-if 3583

This theorem is referenced by:  ifnebibdc  3628