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Theorem cbvcsbv 3906
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 2899 . 2 𝑦𝐵
2 nfcv 2899 . 2 𝑥𝐶
3 cbvcsbv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvcsbw 3904 1 𝐴 / 𝑥𝐵 = 𝐴 / 𝑦𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  csb 3894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-12 2166  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-sbc 3779  df-csb 3895
This theorem is referenced by:  pmatcollpw3lem  22705  precsexlemcbv  28124  poimirlem27  37153  cdleme40v  39974
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