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Theorem cbviunvg 4979
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 2370. Usage of the weaker cbviunv 4977 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 2905 . 2 𝑦𝐵
2 nfcv 2905 . 2 𝑥𝐶
3 cbviunvg.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbviung 4975 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539   ciun 4931
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-13 2370  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-iun 4933
This theorem is referenced by: (None)
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