Theorem necon4ddc 2277

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 2206	. . . 4 |- ( A =/= B <-> -. A = B )
3		df-ne 2206	. . . 4 |- ( C =/= D <-> -. C = D )
4	2, 3	imbi12i 228	. . 3 |- ( ( A =/= B -> C =/= D ) <-> ( -. A = B -> -. C = D ) )
5	1, 4	syl6ib 150	. 2 |- ( ph -> (DECID A = B -> ( -. A = B -> -. C = D ) ) )
6		condc 749	. 2 |- (DECID A = B -> ( ( -. A = B -> -. C = D ) -> ( C = D -> A = B ) ) )
7	5, 6	sylcom 25	1 |- ( ph -> (DECID A = B -> ( C = D -> A = B ) ) )

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