Theorem neeq12d 2329

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 2327	. 2 |- ( ph -> ( A =/= C <-> B =/= C ) )
3		neeq12d.2	. . 3 |- ( ph -> C = D )
4	3	neeq2d 2328	. 2 |- ( ph -> ( B =/= C <-> B =/= D ) )
5	2, 4	bitrd 187	1 |- ( ph -> ( A =/= C <-> B =/= D ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   =/= wne 2309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-ne 2310

This theorem is referenced by:  3netr3d  2341  3netr4d  2342  ennnfonelemim  11973  ctinfom  11977  apdiff  13416