Theorem 3netr4d 2278

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 2265	. 2 |- ( ph -> ( C =/= D <-> A =/= B ) )
5	1, 4	mpbird 165	1 |- ( ph -> C =/= D )

Syntax hints:   -> wi 4   = wceq 1284   =/= wne 2245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-ne 2246

This theorem is referenced by:  modsumfzodifsn  9398