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Theorem cbviunv 3666
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 x A B = y A 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 2156 . 2 yB
2 nfcv 2156 . 2 x𝐶
3 cbviunv.1 . 2 (x = yB = 𝐶)
41, 2, 3cbviun 3664 1 x A B = y A 𝐶
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1226   ciun 3627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-iun 3629
This theorem is referenced by:  iunxdif2  3675
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