Theorem necon4bbiddc 2382

Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)

Hypothesis

Ref	Expression
necon4bbiddc.1	|- ( ph -> (DECID ps -> (DECID A = B -> ( -. ps <-> A =/= B ) ) ) )

Assertion

Ref	Expression
necon4bbiddc	|- ( ph -> (DECID ps -> (DECID A = B -> ( ps <-> A = B ) ) ) )

Proof of Theorem necon4bbiddc

Step	Hyp	Ref	Expression
1		necon4bbiddc.1	. . . . . 6 |- ( ph -> (DECID ps -> (DECID A = B -> ( -. ps <-> A =/= B ) ) ) )
2		bicom 139	. . . . . 6 |- ( ( -. ps <-> A =/= B ) <-> ( A =/= B <-> -. ps ) )
3	1, 2	syl8ib 165	. . . . 5 |- ( ph -> (DECID ps -> (DECID A = B -> ( A =/= B <-> -. ps ) ) ) )
4	3	com23 78	. . . 4 |- ( ph -> (DECID A = B -> (DECID ps -> ( A =/= B <-> -. ps ) ) ) )
5	4	necon4abiddc 2381	. . 3 |- ( ph -> (DECID A = B -> (DECID ps -> ( A = B <-> ps ) ) ) )
6	5	com23 78	. 2 |- ( ph -> (DECID ps -> (DECID A = B -> ( A = B <-> ps ) ) ) )
7		bicom 139	. 2 |- ( ( A = B <-> ps ) <-> ( ps <-> A = B ) )
8	6, 7	syl8ib 165	1 |- ( ph -> (DECID ps -> (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: (None)