Theorem 3netr3d 2237

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

Hypotheses

Ref	Expression
3netr3d.1	|- ( ph -> A =/= B )
3netr3d.2	|- ( ph -> A = C )
3netr3d.3	|- ( ph -> B = D )

Assertion

Ref	Expression
3netr3d	|- ( ph -> C =/= D )

Proof of Theorem 3netr3d

Step	Hyp	Ref	Expression
1		3netr3d.1	. 2 |- ( ph -> A =/= B )
2		3netr3d.2	. . 3 |- ( ph -> A = C )
3		3netr3d.3	. . 3 |- ( ph -> B = D )
4	2, 3	neeq12d 2225	. 2 |- ( ph -> ( A =/= B <-> C =/= D ) )
5	1, 4	mpbid 135	1 |- ( ph -> C =/= D )

Syntax hints:   -> wi 4   = 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: (None)