Theorem necon4ddc 2413

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

Hypothesis

Ref	Expression
necon4ddc.1	|- ( ph -> (DECID A = B -> ( A =/= B -> C =/= D ) ) )

Assertion

Ref	Expression
necon4ddc	|- ( ph -> (DECID A = B -> ( C = D -> A = B ) ) )

Proof of Theorem necon4ddc

Step	Hyp	Ref	Expression
1		necon4ddc.1	. . 3 |- ( ph -> (DECID A = B -> ( A =/= B -> C =/= D ) ) )
2		df-ne 2342	. . . 4 |- ( A =/= B <-> -. A = B )
3		df-ne 2342	. . . 4 |- ( C =/= D <-> -. C = D )
4	2, 3	imbi12i 238	. . 3 |- ( ( A =/= B -> C =/= D ) <-> ( -. A = B -> -. C = D ) )
5	1, 4	syl6ib 160	. 2 |- ( ph -> (DECID A = B -> ( -. A = B -> -. C = D ) ) )
6		condc 849	. 2 |- (DECID A = B -> ( ( -. A = B -> -. C = D ) -> ( C = D -> A = B ) ) )
7	5, 6	sylcom 28	1 |- ( ph -> (DECID A = B -> ( C = D -> A = B ) ) )

Syntax hints:   -. wn 3   -> wi 4  DECID wdc 830   = wceq 1349   =/= wne 2341

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 610  ax-in2 611  ax-io 705

This theorem depends on definitions:  df-bi 116  df-stab 827  df-dc 831  df-ne 2342

This theorem is referenced by: (None)