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Theorem cbvcsbv 2914
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 2220 . 2 𝑦𝐵
2 nfcv 2220 . 2 𝑥𝐶
3 cbvcsbv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvcsb 2913 1 𝐴 / 𝑥𝐵 = 𝐴 / 𝑦𝐶
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  csb 2909
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-sbc 2817  df-csb 2910
This theorem is referenced by: (None)
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