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

Theorem sbceq2a 3732
 Description: Equality theorem for class substitution. Class version of sbequ12r 2251. (Contributed by NM, 4-Jan-2017.)
Assertion
Ref Expression
sbceq2a (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑𝜑))

Proof of Theorem sbceq2a
StepHypRef Expression
1 sbceq1a 3731 . . 3 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
21eqcoms 2806 . 2 (𝐴 = 𝑥 → (𝜑[𝐴 / 𝑥]𝜑))
32bicomd 226 1 (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  [wsbc 3720 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-sbc 3721 This theorem is referenced by:  ralrnmptw  6837  tfindes  7557  rabssnn0fi  13349  indexa  35168  fdc  35180  fdc1  35181  alrimii  35554  tratrbVD  41562
 Copyright terms: Public domain W3C validator