Theorem necon1bbiddc 2410

Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)

Hypothesis

Ref	Expression
necon1bbiddc.1	|- ( ph -> (DECID A = B -> ( A =/= B <-> ps ) ) )

Assertion

Ref	Expression
necon1bbiddc	|- ( ph -> (DECID A = B -> ( -. ps <-> A = B ) ) )

Proof of Theorem necon1bbiddc

Step	Hyp	Ref	Expression
1		necon1bbiddc.1	. . 3 |- ( ph -> (DECID A = B -> ( A =/= B <-> ps ) ) )
2		df-ne 2348	. . . 4 |- ( A =/= B <-> -. A = B )
3	2	bibi1i 228	. . 3 |- ( ( A =/= B <-> ps ) <-> ( -. A = B <-> ps ) )
4	1, 3	syl6ib 161	. 2 |- ( ph -> (DECID A = B -> ( -. A = B <-> ps ) ) )
5	4	con1biddc 876	1 |- ( ph -> (DECID A = B -> ( -. ps <-> A = B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105  DECID wdc 834   = wceq 1353   =/= wne 2347

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 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-ne 2348

This theorem is referenced by:  necon2bbiddc  2414