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

Theorem sbceq2a 3480
Description: Equality theorem for class substitution. Class version of sbequ12r 2150. (Contributed by NM, 4-Jan-2017.)
Assertion
Ref Expression
sbceq2a (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑𝜑))

Proof of Theorem sbceq2a
StepHypRef Expression
1 sbceq1a 3479 . . 3 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
21eqcoms 2659 . 2 (𝐴 = 𝑥 → (𝜑[𝐴 / 𝑥]𝜑))
32bicomd 213 1 (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-sbc 3469
This theorem is referenced by:  tfindes  7104  rabssnn0fi  12825  indexa  33658  fdc  33671  fdc1  33672  alrimii  34054  tratrbVD  39411
  Copyright terms: Public domain W3C validator