Theorem necon2bbiddc 2272

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 128	. . . 4 |- ( ( ps <-> A =/= B ) <-> ( A =/= B <-> ps ) )
3	1, 2	syl6ib 150	. . 3 |- ( ph -> (DECID A = B -> ( A =/= B <-> ps ) ) )
4	3	necon1bbiddc 2268	. 2 |- ( ph -> (DECID A = B -> ( -. ps <-> A = B ) ) )
5		bicom 128	. 2 |- ( ( -. ps <-> A = B ) <-> ( A = B <-> -. ps ) )
6	4, 5	syl6ib 150	1 |- ( ph -> (DECID A = B -> ( A = B <-> -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98  DECID wdc 742   = wceq 1243   =/= wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743  df-ne 2206

This theorem is referenced by: (None)