Theorem ifnetruedc 3614

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 1000	. 2 |- ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) -> DECID ph )
2		iffalse 3580	. . . 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 1005	. . . . 5 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> if ( ph , A , B ) = A )
5		simpl2 1004	. . . . 5 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> A =/= B )
6	4, 5	eqnetrd 2401	. . . 4 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> if ( ph , A , B ) =/= B )
7	6	neneqd 2398	. . 3 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> -. if ( ph , A , B ) = B )
8	3, 7	condandc 883	. 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 836   /\ w3a 981   = wceq 1373   =/= wne 2377   ifcif 3572

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-ne 2378  df-if 3573

This theorem is referenced by:  ifnebibdc  3616