Theorem 3netr4g 2546

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)

Hypotheses

Ref	Expression
3netr4g.1	|- (ph -> A =/= B)
3netr4g.2	|- C = A
3netr4g.3	|- D = B

Assertion

Ref	Expression
3netr4g	|- (ph -> C =/= D)

Proof of Theorem 3netr4g

Step	Hyp	Ref	Expression
1		3netr4g.1	. 2 |- (ph -> A =/= B)
2		3netr4g.2	. . 3 |- C = A
3		3netr4g.3	. . 3 |- D = B
4	2, 3	neeq12i 2529	. 2 |- (C =/= D <-> A =/= B)
5	1, 4	sylibr 203	1 |- (ph -> C =/= D)

Syntax hints:   -> wi 4   = wceq 1642   =/= wne 2517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346  df-ne 2519

This theorem is referenced by: (None)