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Theorem cbviunvg 5047
Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. Usage of this theorem is discouraged because it depends on ax-13 2375. Usage of the weaker cbviunv 5045 is preferred. (Contributed by NM, 15-Sep-2003.) (New usage is discouraged.)
Hypothesis
Ref Expression
cbviunvg.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbviunvg 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 cbviunvg
StepHypRef Expression
1 nfcv 2903 . 2 𝑦𝐵
2 nfcv 2903 . 2 𝑥𝐶
3 cbviunvg.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbviung 5043 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537   ciun 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-iun 4998
This theorem is referenced by: (None)
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