Theorem neeq2 2525

Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Assertion

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

Proof of Theorem neeq2

Step	Hyp	Ref	Expression
1		eqeq2 2362	. . 3 |- (A = B -> (C = A <-> C = B))
2	1	notbid 285	. 2 |- (A = B -> (-. C = A <-> -. C = B))
3		df-ne 2518	. 2 |- (C =/= A <-> -. C = A)
4		df-ne 2518	. 2 |- (C =/= B <-> -. C = B)
5	2, 3, 4	3bitr4g 279	1 |- (A = B -> (C =/= A <-> C =/= B))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   = wceq 1642   =/= wne 2516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346  df-ne 2518

This theorem is referenced by:  neeq2i  2527  neeq2d  2530  psseq2  3357  nfunv  5138  enprmapc  6083  ce2  6192