Theorem necon4abiddc 2278

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 2206	. . . 4 |- ( A =/= B <-> -. A = B )
3	2	bibi1i 217	. . 3 |- ( ( A =/= B <-> -. ps ) <-> ( -. A = B <-> -. ps ) )
4	1, 3	syl8ib 155	. 2 |- ( ph -> (DECID A = B -> (DECID ps -> ( -. A = B <-> -. ps ) ) ) )
5	4	con4biddc 754	1 |- ( ph -> (DECID A = B -> (DECID ps -> ( 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-in2 545  ax-io 630

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

This theorem is referenced by:  necon4bbiddc  2279  necon4biddc  2280