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Theorem cbviun 3694
 Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
Hypotheses
Ref Expression
cbviun.1
cbviun.2
cbviun.3
Assertion
Ref Expression
cbviun
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   (,)

Proof of Theorem cbviun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cbviun.1 . . . . 5
21nfcri 2172 . . . 4
3 cbviun.2 . . . . 5
43nfcri 2172 . . . 4
5 cbviun.3 . . . . 5
65eleq2d 2107 . . . 4
72, 4, 6cbvrex 2530 . . 3
87abbii 2153 . 2
9 df-iun 3659 . 2
10 df-iun 3659 . 2
118, 9, 103eqtr4i 2070 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1243   wcel 1393  cab 2026  wnfc 2165  wrex 2307  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:  cbviunv  3696  funiunfvdmf  5403  mpt2mptsx  5823  dmmpt2ssx  5825  fmpt2x  5826
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