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Theorem sbab 2737
 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 2097 . 2 (𝑥 = 𝑦 → (𝑧𝐴 ↔ [𝑦 / 𝑥]𝑧𝐴))
21abbi2dv 2729 1 (𝑥 = 𝑦𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧𝐴})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  [wsb 1867   ∈ wcel 1977  {cab 2596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606 This theorem is referenced by:  sbcel12  3935  sbceqg  3936  sbcel12gOLD  37775
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