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Theorem cbvcsbv 2938
Description: Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypothesis
Ref Expression
cbvcsbv.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvcsbv 𝐴 / 𝑥𝐵 = 𝐴 / 𝑦𝐶
Distinct variable groups:   𝑥,𝑦   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvcsbv
StepHypRef Expression
1 nfcv 2228 . 2 𝑦𝐵
2 nfcv 2228 . 2 𝑥𝐶
3 cbvcsbv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvcsb 2937 1 𝐴 / 𝑥𝐵 = 𝐴 / 𝑦𝐶
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  csb 2933
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 2841  df-csb 2934
This theorem is referenced by: (None)
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