Theorem neeq2 1591

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 1484	. . 3 |- (A = B -> (C = A <-> C = B))
2	1	negbid 611	. 2 |- (A = B -> (-. C = A <-> -. C = B))
3		df-ne 1587	. 2 |- (C =/= A <-> -. C = A)
4		df-ne 1587	. 2 |- (C =/= B <-> -. C = B)
5	2, 3, 4	3bitr4g 555	1 |- (A = B -> (C =/= A <-> C =/= B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 956   =/= wne 1585

This theorem is referenced by:  neeq2i 1593  neeq2d 1595  psseq2 2136  aceq5 4740  kmlem4 4768  kmlem14 4778  hausnei 7784  superpos 10281  fiiu2 10488  fiiu2OLD 10489  cnfilca 10583  cnfilcaOLD 10584

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-cleq 1469  df-ne 1587

Copyright terms: Public domain