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Theorem cbviun 3658
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 yB
cbviun.2 x𝐶
cbviun.3 (x = yB = 𝐶)
Assertion
Ref Expression
cbviun x A B = y A 𝐶
Distinct variable groups:   y,A   x,A
Allowed substitution hints:   B(x,y)   𝐶(x,y)

Proof of Theorem cbviun
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 cbviun.1 . . . . 5 yB
21nfcri 2146 . . . 4 y z B
3 cbviun.2 . . . . 5 x𝐶
43nfcri 2146 . . . 4 x z 𝐶
5 cbviun.3 . . . . 5 (x = yB = 𝐶)
65eleq2d 2081 . . . 4 (x = y → (z Bz 𝐶))
72, 4, 6cbvrex 2500 . . 3 (x A z By A z 𝐶)
87abbii 2127 . 2 {zx A z B} = {zy A z 𝐶}
9 df-iun 3623 . 2 x A B = {zx A z B}
10 df-iun 3623 . 2 y A 𝐶 = {zy A z 𝐶}
118, 9, 103eqtr4i 2044 1 x A B = y A 𝐶
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1224   wcel 1367  {cab 2000  wnfc 2139  wrex 2277   ciun 3621
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 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996
This theorem depends on definitions:  df-bi 110  df-tru 1227  df-nf 1324  df-sb 1620  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-rex 2282  df-iun 3623
This theorem is referenced by:  cbviunv  3660  funiunfvdmf  5317  mpt2mptsx  5735  dmmpt2ssx  5737  fmpt2x  5738
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