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Theorem cbvcsb 2935
Description: Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
cbvcsb.1 𝑦𝐶
cbvcsb.2 𝑥𝐷
cbvcsb.3 (𝑥 = 𝑦𝐶 = 𝐷)
Assertion
Ref Expression
cbvcsb 𝐴 / 𝑥𝐶 = 𝐴 / 𝑦𝐷

Proof of Theorem cbvcsb
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvcsb.1 . . . . 5 𝑦𝐶
21nfcri 2222 . . . 4 𝑦 𝑧𝐶
3 cbvcsb.2 . . . . 5 𝑥𝐷
43nfcri 2222 . . . 4 𝑥 𝑧𝐷
5 cbvcsb.3 . . . . 5 (𝑥 = 𝑦𝐶 = 𝐷)
65eleq2d 2157 . . . 4 (𝑥 = 𝑦 → (𝑧𝐶𝑧𝐷))
72, 4, 6cbvsbc 2865 . . 3 ([𝐴 / 𝑥]𝑧𝐶[𝐴 / 𝑦]𝑧𝐷)
87abbii 2203 . 2 {𝑧[𝐴 / 𝑥]𝑧𝐶} = {𝑧[𝐴 / 𝑦]𝑧𝐷}
9 df-csb 2932 . 2 𝐴 / 𝑥𝐶 = {𝑧[𝐴 / 𝑥]𝑧𝐶}
10 df-csb 2932 . 2 𝐴 / 𝑦𝐷 = {𝑧[𝐴 / 𝑦]𝑧𝐷}
118, 9, 103eqtr4i 2118 1 𝐴 / 𝑥𝐶 = 𝐴 / 𝑦𝐷
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438  {cab 2074  wnfc 2215  [wsbc 2838  csb 2931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-sbc 2839  df-csb 2932
This theorem is referenced by:  cbvcsbv  2936  cbvsum  10713
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