Theorem ifnetruedc 3598

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 999	. 2 |- ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) -> DECID ph )
2		iffalse 3565	. . . 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 1004	. . . . 5 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> if ( ph , A , B ) = A )
5		simpl2 1003	. . . . 5 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> A =/= B )
6	4, 5	eqnetrd 2388	. . . 4 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> if ( ph , A , B ) =/= B )
7	6	neneqd 2385	. . 3 |- ( ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) /\ -. ph ) -> -. if ( ph , A , B ) = B )
8	3, 7	condandc 882	. 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 835   /\ w3a 980   = wceq 1364   =/= wne 2364   ifcif 3557

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-ne 2365  df-if 3558

This theorem is referenced by:  ifnebibdc  3600