Theorem neeq2 1594

Description: Equality theorem for inequality.

Assertion

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

Proof of Theorem neeq2

Step	Hyp	Ref	Expression
1		eqeq2 1487	. . 3 |- (A = B -> (C = A <-> C = B))
2	1	negbid 613	. 2 |- (A = B -> (-. C = A <-> -. C = B))
3		df-ne 1590	. 2 |- (C =/= A <-> -. C = A)
4		df-ne 1590	. 2 |- (C =/= B <-> -. C = B)
5	2, 3, 4	3bitr4g 557	1 |- (A = B -> (C =/= A <-> C =/= B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 958   =/= wne 1588

This theorem is referenced by:  neeq2i 1596  neeq2d 1598  psseq2 2139  aceq5 4750  kmlem4 4778  kmlem14 4788  hausnei 7781  superpos 10276  fiiu2 10473  cnfilca 10562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-cleq 1472  df-ne 1590

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