Theorem necon2bbiddc 2322

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

Hypothesis

Ref	Expression
necon2bbiddc.1	|- ( ph -> (DECID A = B -> ( ps <-> A =/= B ) ) )

Assertion

Ref	Expression
necon2bbiddc	|- ( ph -> (DECID A = B -> ( A = B <-> -. ps ) ) )

Proof of Theorem necon2bbiddc

Step	Hyp	Ref	Expression
1		necon2bbiddc.1	. . . 4 |- ( ph -> (DECID A = B -> ( ps <-> A =/= B ) ) )
2		bicom 138	. . . 4 |- ( ( ps <-> A =/= B ) <-> ( A =/= B <-> ps ) )
3	1, 2	syl6ib 159	. . 3 |- ( ph -> (DECID A = B -> ( A =/= B <-> ps ) ) )
4	3	necon1bbiddc 2318	. 2 |- ( ph -> (DECID A = B -> ( -. ps <-> A = B ) ) )
5		bicom 138	. 2 |- ( ( -. ps <-> A = B ) <-> ( A = B <-> -. ps ) )
6	4, 5	syl6ib 159	1 |- ( ph -> (DECID A = B -> ( A = B <-> -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103  DECID wdc 780   = wceq 1289   =/= wne 2255

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665

This theorem depends on definitions:  df-bi 115  df-dc 781  df-ne 2256

This theorem is referenced by: (None)