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Theorem eqsbc1 3764
Description: Substitution for the left-hand side in an equality. Class version of eqsb1 2865. (Contributed by Andrew Salmon, 29-Jun-2011.) Avoid ax-13 2372. (Revised by Wolf Lammen, 29-Apr-2023.)
Assertion
Ref Expression
eqsbc1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem eqsbc1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3717 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 = 𝐵[𝐴 / 𝑥]𝑥 = 𝐵))
2 eqeq1 2742 . 2 (𝑦 = 𝐴 → (𝑦 = 𝐵𝐴 = 𝐵))
3 sbsbc 3719 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐵[𝑦 / 𝑥]𝑥 = 𝐵)
4 eqsb1 2865 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐵𝑦 = 𝐵)
53, 4bitr3i 276 . 2 ([𝑦 / 𝑥]𝑥 = 𝐵𝑦 = 𝐵)
61, 2, 5vtoclbg 3504 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  [wsb 2067  wcel 2106  [wsbc 3715
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-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-sbc 3716
This theorem is referenced by:  sbceqalOLD  3782  eqsbc2  3784  fmptsnd  7033  fvmptnn04if  22008  snfil  23025  f1omptsnlem  35515  mptsnunlem  35517  topdifinffinlem  35526  relowlpssretop  35543  iotavalb  42029  onfrALTlem5  42143  eqsbc2VD  42441  onfrALTlem5VD  42486
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