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Theorem sbab 2908
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 2180 . 2 (𝑥 = 𝑦 → (𝑧𝐴 ↔ [𝑦 / 𝑥]𝑧𝐴))
21abbi2dv 2895 1 (𝑥 = 𝑦𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1508  [wsb 2016  wcel 2051  {cab 2751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-12 2107  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839
This theorem is referenced by:  sbcel12  4240  sbceqg  4241
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