Theorem necon1bbiddc 2371

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 2309	. . . 4 |- ( A =/= B <-> -. A = B )
3	2	bibi1i 227	. . 3 |- ( ( A =/= B <-> ps ) <-> ( -. A = B <-> ps ) )
4	1, 3	syl6ib 160	. 2 |- ( ph -> (DECID A = B -> ( -. A = B <-> ps ) ) )
5	4	con1biddc 861	1 |- ( ph -> (DECID A = B -> ( -. ps <-> A = B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104  DECID wdc 819   = wceq 1331   =/= wne 2308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-ne 2309

This theorem is referenced by:  necon2bbiddc  2375