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Theorem sbab 2317
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 1782 . 2 (𝑥 = 𝑦 → (𝑧𝐴 ↔ [𝑦 / 𝑥]𝑧𝐴))
21abbi2dv 2308 1 (𝑥 = 𝑦𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  [wsb 1773  wcel 2160  {cab 2175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185
This theorem is referenced by:  sbcel12g  3087  sbceqg  3088
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