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Theorem csbeq2d 2997
Description: Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
csbeq2d.1 𝑥𝜑
csbeq2d.2 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
csbeq2d (𝜑𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)

Proof of Theorem csbeq2d
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq2d.1 . . . 4 𝑥𝜑
2 csbeq2d.2 . . . . 5 (𝜑𝐵 = 𝐶)
32eleq2d 2187 . . . 4 (𝜑 → (𝑦𝐵𝑦𝐶))
41, 3sbcbid 2938 . . 3 (𝜑 → ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2235 . 2 (𝜑 → {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
6 df-csb 2976 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
7 df-csb 2976 . 2 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
85, 6, 73eqtr4g 2175 1 (𝜑𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wnf 1421  wcel 1465  {cab 2103  [wsbc 2882  csb 2975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-sbc 2883  df-csb 2976
This theorem is referenced by:  csbeq2dv  2998
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