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

Theorem 3netr4g 3099
 Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
Hypotheses
Ref Expression
3netr4g.1 (𝜑𝐴𝐵)
3netr4g.2 𝐶 = 𝐴
3netr4g.3 𝐷 = 𝐵
Assertion
Ref Expression
3netr4g (𝜑𝐶𝐷)

Proof of Theorem 3netr4g
StepHypRef Expression
1 3netr4g.1 . 2 (𝜑𝐴𝐵)
2 3netr4g.2 . . 3 𝐶 = 𝐴
3 3netr4g.3 . . 3 𝐷 = 𝐵
42, 3neeq12i 3086 . 2 (𝐶𝐷𝐴𝐵)
51, 4sylibr 235 1 (𝜑𝐶𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1530   ≠ wne 3020 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-9 2116  ax-ext 2796 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-cleq 2817  df-ne 3021 This theorem is referenced by:  aalioulem2  24837  mapdpglem18  38693  line2x  44570
 Copyright terms: Public domain W3C validator