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Theorem cbviun 3668
 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 2154 . . . 4 y z B
3 cbviun.2 . . . . 5 x𝐶
43nfcri 2154 . . . 4 x z 𝐶
5 cbviun.3 . . . . 5 (x = yB = 𝐶)
65eleq2d 2089 . . . 4 (x = y → (z Bz 𝐶))
72, 4, 6cbvrex 2508 . . 3 (x A z By A z 𝐶)
87abbii 2135 . 2 {zx A z B} = {zy A z 𝐶}
9 df-iun 3633 . 2 x A B = {zx A z B}
10 df-iun 3633 . 2 y A 𝐶 = {zy A z 𝐶}
118, 9, 103eqtr4i 2052 1 x A B = y A 𝐶
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1228   ∈ wcel 1374  {cab 2008  Ⅎwnfc 2147  ∃wrex 2285  ∪ ciun 3631 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-iun 3633 This theorem is referenced by:  cbviunv  3670  funiunfvdmf  5328  mpt2mptsx  5746  dmmpt2ssx  5748  fmpt2x  5749
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