MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3netr3g Unicode version

Theorem 3netr3g 2474
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
StepHypRef Expression
1 3netr3g.1 . 2  |-  ( ph  ->  A  =/=  B )
2 3netr3g.2 . . 3  |-  A  =  C
3 3netr3g.3 . . 3  |-  B  =  D
42, 3neeq12i 2458 . 2  |-  ( A  =/=  B  <->  C  =/=  D )
51, 4sylib 188 1  |-  ( ph  ->  C  =/=  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    =/= wne 2446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-cleq 2276  df-ne 2448
  Copyright terms: Public domain W3C validator