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Theorem cbviunv 3822
Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)
Hypothesis
Ref Expression
cbviunv.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbviunv 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbviunv
StepHypRef Expression
1 nfcv 2258 . 2 𝑦𝐵
2 nfcv 2258 . 2 𝑥𝐶
3 cbviunv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbviun 3820 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316   ciun 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-iun 3785
This theorem is referenced by:  iunxdif2  3831  ennnfonelemnn0  11862
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