MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqsbc3 Structured version   Visualization version   GIF version

Theorem eqsbc3 3768
Description: Substitution applied to an atomic wff. Class version of eqsb3 2919. (Contributed by Andrew Salmon, 29-Jun-2011.) Avoid ax-13 2382. (Revised by Wolf Lammen, 29-Apr-2023.)
Assertion
Ref Expression
eqsbc3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem eqsbc3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3725 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 = 𝐵[𝐴 / 𝑥]𝑥 = 𝐵))
2 eqeq1 2805 . 2 (𝑦 = 𝐴 → (𝑦 = 𝐵𝐴 = 𝐵))
3 sbsbc 3727 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐵[𝑦 / 𝑥]𝑥 = 𝐵)
4 eqsb3 2919 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐵𝑦 = 𝐵)
53, 4bitr3i 280 . 2 ([𝑦 / 𝑥]𝑥 = 𝐵𝑦 = 𝐵)
61, 2, 5vtoclbg 3520 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  [wsb 2069  wcel 2112  [wsbc 3723
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 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-sbc 3724
This theorem is referenced by:  sbceqal  3785  eqsbc3r  3787  fmptsnd  6912  fvmptnn04if  21457  snfil  22472  f1omptsnlem  34748  mptsnunlem  34750  topdifinffinlem  34759  relowlpssretop  34776  iotavalb  41121  onfrALTlem5  41235  eqsbc3rVD  41533  onfrALTlem5VD  41578
  Copyright terms: Public domain W3C validator