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Theorem cbviunv 3696
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  =  y  ->  B  =  C )
Assertion
Ref Expression
cbviunv  |-  U_ x  e.  A  B  =  U_ y  e.  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 2178 . 2  |-  F/_ y B
2 nfcv 2178 . 2  |-  F/_ x C
3 cbviunv.1 . 2  |-  ( x  =  y  ->  B  =  C )
41, 2, 3cbviun 3694 1  |-  U_ x  e.  A  B  =  U_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   U_ciun 3657
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-iun 3659
This theorem is referenced by:  iunxdif2  3705
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