Theorem 3netr4d 2369

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Hypotheses

Ref	Expression
3netr4d.1	|- ( ph -> A =/= B )
3netr4d.2	|- ( ph -> C = A )
3netr4d.3	|- ( ph -> D = B )

Assertion

Ref	Expression
3netr4d	|- ( ph -> C =/= D )

Proof of Theorem 3netr4d

Step	Hyp	Ref	Expression
1		3netr4d.1	. 2 |- ( ph -> A =/= B )
2		3netr4d.2	. . 3 |- ( ph -> C = A )
3		3netr4d.3	. . 3 |- ( ph -> D = B )
4	2, 3	neeq12d 2356	. 2 |- ( ph -> ( C =/= D <-> A =/= B ) )
5	1, 4	mpbird 166	1 |- ( ph -> C =/= D )

Syntax hints:   -> wi 4   = wceq 1343   =/= wne 2336

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 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-ne 2337

This theorem is referenced by:  modsumfzodifsn  10331  ennnfonelemnn0  12355  lgsne0  13579