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Theorem sbab 2161
 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]z A})
Distinct variable groups:   z,A   x,z   y,z
Allowed substitution hints:   A(x,y)

Proof of Theorem sbab
StepHypRef Expression
1 sbequ12 1651 . 2 (x = y → (z A ↔ [y / x]z A))
21abbi2dv 2153 1 (x = yA = {z ∣ [y / x]z A})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  [wsb 1642  {cab 2023 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033 This theorem is referenced by:  sbcel12g  2859  sbceqg  2860
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