MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difeq12 Structured version   Visualization version   GIF version

Theorem difeq12 4052
Description: Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
Assertion
Ref Expression
difeq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem difeq12
StepHypRef Expression
1 difeq1 4050 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 difeq2 4051 . 2 (𝐶 = 𝐷 → (𝐵𝐶) = (𝐵𝐷))
31, 2sylan9eq 2798 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  cdif 3884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-dif 3890
This theorem is referenced by:  resdif  6737
  Copyright terms: Public domain W3C validator