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Theorem cbviinv 4957
Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) Add disjoint variable condition to avoid ax-13 2381. See cbviinvg 4959 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 2974 . 2 𝑦𝐵
2 nfcv 2974 . 2 𝑥𝐶
3 cbviunv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbviin 4953 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528   ciin 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-iin 4913
This theorem is referenced by:  meaiininc  42646  iinhoiicc  42833  smflimlem3  42926  smflimlem4  42927  smflimlem6  42929  smfsuplem2  42963  smflimsuplem1  42971  smflimsup  42979
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