Theorem necon1ddc 2414

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

Hypothesis

Ref	Expression
necon1ddc.1	|- ( ph -> (DECID A = B -> ( A =/= B -> C = D ) ) )

Assertion

Ref	Expression
necon1ddc	|- ( ph -> (DECID A = B -> ( C =/= D -> A = B ) ) )

Proof of Theorem necon1ddc

Step	Hyp	Ref	Expression
1		df-ne 2337	. 2 |- ( C =/= D <-> -. C = D )
2		necon1ddc.1	. . 3 |- ( ph -> (DECID A = B -> ( A =/= B -> C = D ) ) )
3	2	necon1bddc 2413	. 2 |- ( ph -> (DECID A = B -> ( -. C = D -> A = B ) ) )
4	1, 3	syl7bi 164	1 |- ( ph -> (DECID A = B -> ( C =/= D -> 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:  xblss2ps  13044  xblss2  13045  lgsne0  13579