Theorem 3netr3g 2872

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

Hypotheses

Ref	Expression
3netr3g.1	|- ( ph -> A =/= B )
3netr3g.2	|- A = C
3netr3g.3	|- B = D

Assertion

Ref	Expression
3netr3g	|- ( ph -> C =/= D )

Proof of Theorem 3netr3g

Step	Hyp	Ref	Expression
1		3netr3g.1	. 2 |- ( ph -> A =/= B )
2		3netr3g.2	. . 3 |- A = C
3		3netr3g.3	. . 3 |- B = D
4	2, 3	neeq12i 2860	. 2 |- ( A =/= B <-> C =/= D )
5	1, 4	sylib 208	1 |- ( ph -> C =/= D )

Syntax hints:   -> wi 4   = wceq 1483   =/= wne 2794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-ne 2795

This theorem is referenced by: (None)