Theorem neeq12d 2395

Description: Deduction for inequality. (Contributed by NM, 24-Jul-2012.)

Hypotheses

Ref	Expression
neeq1d.1	|- ( ph -> A = B )
neeq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
neeq12d	|- ( ph -> ( A =/= C <-> B =/= D ) )

Proof of Theorem neeq12d

Step	Hyp	Ref	Expression
1		neeq1d.1	. . 3 |- ( ph -> A = B )
2	1	neeq1d 2393	. 2 |- ( ph -> ( A =/= C <-> B =/= C ) )
3		neeq12d.2	. . 3 |- ( ph -> C = D )
4	3	neeq2d 2394	. 2 |- ( ph -> ( B =/= C <-> B =/= D ) )
5	2, 4	bitrd 188	1 |- ( ph -> ( A =/= C <-> B =/= D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1372   =/= wne 2375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1469  ax-gen 1471  ax-4 1532  ax-17 1548  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-cleq 2197  df-ne 2376

This theorem is referenced by:  3netr3d  2407  3netr4d  2408  exmidapne  7371  ennnfonelemim  12766  ctinfom  12770  isnzr  13914  opprnzrbg  13918  apdiff  15949