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Theorem cbvcsbv 3505
 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 2751 . 2 𝑦𝐵
2 nfcv 2751 . 2 𝑥𝐶
3 cbvcsbv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvcsb 3504 1 𝐴 / 𝑥𝐵 = 𝐴 / 𝑦𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ⦋csb 3499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-sbc 3403  df-csb 3500 This theorem is referenced by:  pmatcollpw3lem  20407  poimirlem27  32606  cdleme40v  34775
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