Theorem necon4bddc 2407

Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)

Hypothesis

Ref	Expression
necon4bddc.1	|- ( ph -> (DECID ps -> ( -. ps -> A =/= B ) ) )

Assertion

Ref	Expression
necon4bddc	|- ( ph -> (DECID ps -> ( A = B -> ps ) ) )

Proof of Theorem necon4bddc

Step	Hyp	Ref	Expression
1		necon4bddc.1	. . 3 |- ( ph -> (DECID ps -> ( -. ps -> A =/= B ) ) )
2		df-ne 2337	. . 3 |- ( A =/= B <-> -. A = B )
3	1, 2	syl8ib 165	. 2 |- ( ph -> (DECID ps -> ( -. ps -> -. A = B ) ) )
4		condc 843	. 2 |- (DECID ps -> ( ( -. ps -> -. A = B ) -> ( A = B -> ps ) ) )
5	3, 4	sylcom 28	1 |- ( ph -> (DECID ps -> ( A = B -> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4  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: (None)