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Theorem cbvcsbv 3911
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
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvcsbv.1 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
21eleq2d 2827 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
32cbvsbcvw 3822 . . 3 ([𝐴 / 𝑥]𝑧𝐵[𝐴 / 𝑦]𝑧𝐶)
43abbii 2809 . 2 {𝑧[𝐴 / 𝑥]𝑧𝐵} = {𝑧[𝐴 / 𝑦]𝑧𝐶}
5 df-csb 3900 . 2 𝐴 / 𝑥𝐵 = {𝑧[𝐴 / 𝑥]𝑧𝐵}
6 df-csb 3900 . 2 𝐴 / 𝑦𝐶 = {𝑧[𝐴 / 𝑦]𝑧𝐶}
74, 5, 63eqtr4i 2775 1 𝐴 / 𝑥𝐵 = 𝐴 / 𝑦𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {cab 2714  [wsbc 3788  csb 3899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3789  df-csb 3900
This theorem is referenced by:  cbvsumv  15732  cbvprodv  15950  pmatcollpw3lem  22789  precsexlemcbv  28230  cbvprodvw2  36248  poimirlem27  37654  cdleme40v  40471
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