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Theorem eqsbc3 3814
Description: Substitution applied to an atomic wff. Class version of eqsb3 2936. (Contributed by Andrew Salmon, 29-Jun-2011.) Avoid ax-13 2381. (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 3771 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 = 𝐵[𝐴 / 𝑥]𝑥 = 𝐵))
2 eqeq1 2822 . 2 (𝑦 = 𝐴 → (𝑦 = 𝐵𝐴 = 𝐵))
3 sbsbc 3773 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐵[𝑦 / 𝑥]𝑥 = 𝐵)
4 eqsb3 2936 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐵𝑦 = 𝐵)
53, 4bitr3i 278 . 2 ([𝑦 / 𝑥]𝑥 = 𝐵𝑦 = 𝐵)
61, 2, 5vtoclbg 3566 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  [wsb 2060  wcel 2105  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-sbc 3770
This theorem is referenced by:  sbceqal  3832  eqsbc3r  3834  fmptsnd  6923  fvmptnn04if  21385  snfil  22400  f1omptsnlem  34499  mptsnunlem  34501  topdifinffinlem  34510  relowlpssretop  34527  iotavalb  40639  onfrALTlem5  40753  eqsbc3rVD  41051  onfrALTlem5VD  41096
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