Theorem neeq2 1195

Description: Equality theorem for inequality.

Assertion

Ref	Expression
neeq2	|- (A = B -> (C =/= A <-> C =/= B))

Proof of Theorem neeq2

Step	Hyp	Ref	Expression
1		cleq2 1110	. . 3 |- (A = B -> (C = A <-> C = B))
2	1	negbid 463	. 2 |- (A = B -> (-. C = A <-> -. C = B))
3		df-ne 1192	. 2 |- (C =/= A <-> -. C = A)
4		df-ne 1192	. 2 |- (C =/= B <-> -. C = B)
5	2, 3, 4	3bitr4g 428	1 |- (A = B -> (C =/= A <-> C =/= B))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   = wceq 1091   =/= wne 1190

This theorem is referenced by:  neeq2d 1197  psseq2 1560

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-cleq 1097  df-ne 1192

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