MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3netr4g Structured version   Unicode version

Theorem 3netr4g 2739
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
StepHypRef Expression
1 3netr4g.1 . 2  |-  ( ph  ->  A  =/=  B )
2 3netr4g.2 . . 3  |-  C  =  A
3 3netr4g.3 . . 3  |-  D  =  B
42, 3neeq12i 2720 . 2  |-  ( C  =/=  D  <->  A  =/=  B )
51, 4sylibr 215 1  |-  ( ph  ->  C  =/=  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    =/= wne 2625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-cleq 2421  df-ne 2627
This theorem is referenced by:  aalioulem2  23154  mapdpglem18  34969
  Copyright terms: Public domain W3C validator