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Theorem cbvcsbw 3073
Description: Version of cbvcsb 3074 with a disjoint variable condition. (Contributed by Gino Giotto, 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 2323 . . . 4 𝑦 𝑧𝐶
3 cbvcsbw.2 . . . . 5 𝑥𝐷
43nfcri 2323 . . . 4 𝑥 𝑧𝐷
5 cbvcsbw.3 . . . . 5 (𝑥 = 𝑦𝐶 = 𝐷)
65eleq2d 2257 . . . 4 (𝑥 = 𝑦 → (𝑧𝐶𝑧𝐷))
72, 4, 6cbvsbcw 3002 . . 3 ([𝐴 / 𝑥]𝑧𝐶[𝐴 / 𝑦]𝑧𝐷)
87abbii 2303 . 2 {𝑧[𝐴 / 𝑥]𝑧𝐶} = {𝑧[𝐴 / 𝑦]𝑧𝐷}
9 df-csb 3070 . 2 𝐴 / 𝑥𝐶 = {𝑧[𝐴 / 𝑥]𝑧𝐶}
10 df-csb 3070 . 2 𝐴 / 𝑦𝐷 = {𝑧[𝐴 / 𝑦]𝑧𝐷}
118, 9, 103eqtr4i 2218 1 𝐴 / 𝑥𝐶 = 𝐴 / 𝑦𝐷
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wcel 2158  {cab 2173  wnfc 2316  [wsbc 2974  csb 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-sbc 2975  df-csb 3070
This theorem is referenced by:  cbvprod  11579
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