Theorem neeq1 2218

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

Assertion

Ref	Expression
neeq1	|- ( A = B -> ( A =/= C <-> B =/= C ) )

Proof of Theorem neeq1

Step	Hyp	Ref	Expression
1		eqeq1 2046	. . 3 |- ( A = B -> ( A = C <-> B = C ) )
2	1	notbid 592	. 2 |- ( A = B -> ( -. A = C <-> -. B = C ) )
3		df-ne 2206	. 2 |- ( A =/= C <-> -. A = C )
4		df-ne 2206	. 2 |- ( B =/= C <-> -. B = C )
5	2, 3, 4	3bitr4g 212	1 |- ( A = B -> ( A =/= C <-> B =/= C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   = wceq 1243   =/= wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-ne 2206

This theorem is referenced by:  neeq1i  2220  neeq1d  2223  nelrdva  2746  psseq1  3031  0inp0  3919  uzn0  8488  xrnemnf  8699  xrnepnf  8700  ngtmnft  8731  fztpval  8945