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Theorem cbviun 3685
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 2169 . . . 4 y z B
3 cbviun.2 . . . . 5 x𝐶
43nfcri 2169 . . . 4 x z 𝐶
5 cbviun.3 . . . . 5 (x = yB = 𝐶)
65eleq2d 2104 . . . 4 (x = y → (z Bz 𝐶))
72, 4, 6cbvrex 2524 . . 3 (x A z By A z 𝐶)
87abbii 2150 . 2 {zx A z B} = {zy A z 𝐶}
9 df-iun 3650 . 2 x A B = {zx A z B}
10 df-iun 3650 . 2 y A 𝐶 = {zy A z 𝐶}
118, 9, 103eqtr4i 2067 1 x A B = y A 𝐶
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  {cab 2023  wnfc 2162  wrex 2301   ciun 3648
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-iun 3650
This theorem is referenced by:  cbviunv  3687  funiunfvdmf  5346  mpt2mptsx  5765  dmmpt2ssx  5767  fmpt2x  5768
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