MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbceq2a Structured version   Visualization version   GIF version

Theorem sbceq2a 3785
Description: Equality theorem for class substitution. Class version of sbequ12r 2239. (Contributed by NM, 4-Jan-2017.)
Assertion
Ref Expression
sbceq2a (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑𝜑))

Proof of Theorem sbceq2a
StepHypRef Expression
1 sbceq1a 3784 . . 3 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
21eqcoms 2733 . 2 (𝐴 = 𝑥 → (𝜑[𝐴 / 𝑥]𝜑))
32bicomd 222 1 (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  [wsbc 3773
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-sbc 3774
This theorem is referenced by:  ralrnmptw  7103  tfindes  7868  rabssnn0fi  13992  indexa  37339  fdc  37351  fdc1  37352  alrimii  37725  tratrbVD  44444
  Copyright terms: Public domain W3C validator