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Theorem cbvcsbw 3909
Description: Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. Version of cbvcsb 3910 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Jeff Hankins, 13-Sep-2009.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvcsbw.1 𝑦𝐶
cbvcsbw.2 𝑥𝐷
cbvcsbw.3 (𝑥 = 𝑦𝐶 = 𝐷)
Assertion
Ref Expression
cbvcsbw 𝐴 / 𝑥𝐶 = 𝐴 / 𝑦𝐷
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem cbvcsbw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvcsbw.1 . . . . 5 𝑦𝐶
21nfcri 2897 . . . 4 𝑦 𝑧𝐶
3 cbvcsbw.2 . . . . 5 𝑥𝐷
43nfcri 2897 . . . 4 𝑥 𝑧𝐷
5 cbvcsbw.3 . . . . 5 (𝑥 = 𝑦𝐶 = 𝐷)
65eleq2d 2827 . . . 4 (𝑥 = 𝑦 → (𝑧𝐶𝑧𝐷))
72, 4, 6cbvsbcw 3821 . . 3 ([𝐴 / 𝑥]𝑧𝐶[𝐴 / 𝑦]𝑧𝐷)
87abbii 2809 . 2 {𝑧[𝐴 / 𝑥]𝑧𝐶} = {𝑧[𝐴 / 𝑦]𝑧𝐷}
9 df-csb 3900 . 2 𝐴 / 𝑥𝐶 = {𝑧[𝐴 / 𝑥]𝑧𝐶}
10 df-csb 3900 . 2 𝐴 / 𝑦𝐷 = {𝑧[𝐴 / 𝑦]𝑧𝐷}
118, 9, 103eqtr4i 2775 1 𝐴 / 𝑥𝐶 = 𝐴 / 𝑦𝐷
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {cab 2714  wnfc 2890  [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-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-sbc 3789  df-csb 3900
This theorem is referenced by:  cbvsum  15731  cbvprod  15949  measiuns  34218  poimirlem26  37653  climinf2mpt  45729  climinfmpt  45730
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