Theorem neeq2 1583

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 1476	. . 3 |- (A = B -> (C = A <-> C = B))
2	1	negbid 609	. 2 |- (A = B -> (-. C = A <-> -. C = B))
3		df-ne 1579	. 2 |- (C =/= A <-> -. C = A)
4		df-ne 1579	. 2 |- (C =/= B <-> -. C = B)
5	2, 3, 4	3bitr4g 553	1 |- (A = B -> (C =/= A <-> C =/= B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 953   =/= wne 1577

This theorem is referenced by:  neeq2i 1585  neeq2d 1587  psseq2 2126  aceq5 4712  kmlem4 4740  kmlem14 4750  hausnei 7723  superpos 10189  fiiu2 10377  cnfilca 10451

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-17 968  ax-4 970  ax-5o 972  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-cleq 1462  df-ne 1579

Copyright terms: Public domain