Theorem necon4aidc 2404

Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)

Hypothesis

Ref	Expression
necon4aidc.1	|- (DECID A = B -> ( A =/= B -> -. ph ) )

Assertion

Ref	Expression
necon4aidc	|- (DECID A = B -> ( ph -> A = B ) )

Proof of Theorem necon4aidc

Step	Hyp	Ref	Expression
1		df-ne 2337	. . 3 |- ( A =/= B <-> -. A = B )
2		necon4aidc.1	. . 3 |- (DECID A = B -> ( A =/= B -> -. ph ) )
3	1, 2	syl5bir 152	. 2 |- (DECID A = B -> ( -. A = B -> -. ph ) )
4		condc 843	. 2 |- (DECID A = B -> ( ( -. A = B -> -. ph ) -> ( ph -> A = B ) ) )
5	3, 4	mpd 13	1 |- (DECID A = B -> ( ph -> A = B ) )

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:  necon4idc  2405