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Theorem sbab 2878
Description: The right-hand side of the second equality is a way of representing proper substitution of 𝑦 for 𝑥 into a class variable. (Contributed by NM, 14-Sep-2003.)
Assertion
Ref Expression
sbab (𝑥 = 𝑦𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧𝐴})
Distinct variable groups:   𝑧,𝐴   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem sbab
StepHypRef Expression
1 sbequ12 2239 . 2 (𝑥 = 𝑦 → (𝑧𝐴 ↔ [𝑦 / 𝑥]𝑧𝐴))
21eqabdv 2863 1 (𝑥 = 𝑦𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  [wsb 2060  wcel 2099  {cab 2705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806
This theorem is referenced by:  sbcel12  4404  sbceqg  4405
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