Theorem ifnefals 3627

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

Assertion

Ref	Expression
ifnefals	|- ( ( A =/= B /\ if ( ph , A , B ) = B ) -> -. ph )

Proof of Theorem ifnefals

Step	Hyp	Ref	Expression
1		iftrue 3587	. . 3 |- ( ph -> if ( ph , A , B ) = A )
2	1	adantl 277	. 2 |- ( ( ( A =/= B /\ if ( ph , A , B ) = B ) /\ ph ) -> if ( ph , A , B ) = A )
3		simplr 528	. . . 4 |- ( ( ( A =/= B /\ if ( ph , A , B ) = B ) /\ ph ) -> if ( ph , A , B ) = B )
4		simpll 527	. . . . 5 |- ( ( ( A =/= B /\ if ( ph , A , B ) = B ) /\ ph ) -> A =/= B )
5	4	necomd 2466	. . . 4 |- ( ( ( A =/= B /\ if ( ph , A , B ) = B ) /\ ph ) -> B =/= A )
6	3, 5	eqnetrd 2404	. . 3 |- ( ( ( A =/= B /\ if ( ph , A , B ) = B ) /\ ph ) -> if ( ph , A , B ) =/= A )
7	6	neneqd 2401	. 2 |- ( ( ( A =/= B /\ if ( ph , A , B ) = B ) /\ ph ) -> -. if ( ph , A , B ) = A )
8	2, 7	pm2.65da 665	1 |- ( ( A =/= B /\ if ( ph , A , B ) = B ) -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = 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-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