Theorem necon2bbiddc 2407

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 139	. . . 4 |- ( ( ps <-> A =/= B ) <-> ( A =/= B <-> ps ) )
3	1, 2	syl6ib 160	. . 3 |- ( ph -> (DECID A = B -> ( A =/= B <-> ps ) ) )
4	3	necon1bbiddc 2403	. 2 |- ( ph -> (DECID A = B -> ( -. ps <-> A = B ) ) )
5		bicom 139	. 2 |- ( ( -. ps <-> A = B ) <-> ( A = B <-> -. ps ) )
6	4, 5	syl6ib 160	1 |- ( ph -> (DECID A = B -> ( A = B <-> -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104  DECID wdc 829   = wceq 1348   =/= wne 2340

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 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-ne 2341

This theorem is referenced by: (None)