Theorem necon4abiddc 2409

Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.)

Hypothesis

Ref	Expression
necon4abiddc.1	|- ( ph -> (DECID A = B -> (DECID ps -> ( A =/= B <-> -. ps ) ) ) )

Assertion

Ref	Expression
necon4abiddc	|- ( ph -> (DECID A = B -> (DECID ps -> ( A = B <-> ps ) ) ) )

Proof of Theorem necon4abiddc

Step	Hyp	Ref	Expression
1		necon4abiddc.1	. . 3 |- ( ph -> (DECID A = B -> (DECID ps -> ( A =/= B <-> -. ps ) ) ) )
2		df-ne 2337	. . . 4 |- ( A =/= B <-> -. A = B )
3	2	bibi1i 227	. . 3 |- ( ( A =/= B <-> -. ps ) <-> ( -. A = B <-> -. ps ) )
4	1, 3	syl8ib 165	. 2 |- ( ph -> (DECID A = B -> (DECID ps -> ( -. A = B <-> -. ps ) ) ) )
5	4	con4biddc 847	1 |- ( ph -> (DECID A = B -> (DECID ps -> ( A = B <-> ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104  DECID wdc 824   = wceq 1343   =/= wne 2336

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-ne 2337

This theorem is referenced by:  necon4bbiddc  2410  necon4biddc  2411  lgsprme0  13583