Theorem 3netr3d 2379

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 2367	. 2 |- ( ph -> ( A =/= B <-> C =/= D ) )
5	1, 4	mpbid 147	1 |- ( ph -> C =/= D )

Syntax hints:   -> wi 4   = wceq 1353   =/= wne 2347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-ne 2348

This theorem is referenced by:  subrgnzr  13301