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Theorem cbviinv 4931
Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) Add disjoint variable condition to avoid ax-13 2382. See cbviinvg 4933 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Hypothesis
Ref Expression
cbviunv.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbviinv 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbviinv
StepHypRef Expression
1 nfcv 2958 . 2 𝑦𝐵
2 nfcv 2958 . 2 𝑥𝐶
3 cbviunv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbviin 4927 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538   ciin 4885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-iin 4887
This theorem is referenced by:  meaiininc  43119  iinhoiicc  43306  smflimlem3  43399  smflimlem4  43400  smflimlem6  43402  smfsuplem2  43436  smflimsuplem1  43444  smflimsup  43452
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