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Theorem iinin1m 3953
Description: Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
iinin1m (∃𝑥 𝑥𝐴 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iinin1m
StepHypRef Expression
1 iinin2m 3952 . 2 (∃𝑥 𝑥𝐴 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
2 incom 3327 . . . 4 (𝐶𝐵) = (𝐵𝐶)
32a1i 9 . . 3 (𝑥𝐴 → (𝐶𝐵) = (𝐵𝐶))
43iineq2i 3903 . 2 𝑥𝐴 (𝐶𝐵) = 𝑥𝐴 (𝐵𝐶)
5 incom 3327 . 2 ( 𝑥𝐴 𝐶𝐵) = (𝐵 𝑥𝐴 𝐶)
61, 4, 53eqtr4g 2235 1 (∃𝑥 𝑥𝐴 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wex 1492  wcel 2148  cin 3128   ciin 3885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-in 3135  df-iin 3887
This theorem is referenced by: (None)
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