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Theorem iinin1m 3776
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 3775 . 2 (∃𝑥 𝑥𝐴 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
2 incom 3178 . . . 4 (𝐶𝐵) = (𝐵𝐶)
32a1i 9 . . 3 (𝑥𝐴 → (𝐶𝐵) = (𝐵𝐶))
43iineq2i 3726 . 2 𝑥𝐴 (𝐶𝐵) = 𝑥𝐴 (𝐵𝐶)
5 incom 3178 . 2 ( 𝑥𝐴 𝐶𝐵) = (𝐵 𝑥𝐴 𝐶)
61, 4, 53eqtr4g 2142 1 (∃𝑥 𝑥𝐴 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wex 1424  wcel 1436  cin 2985   ciin 3708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2616  df-in 2992  df-iin 3710
This theorem is referenced by: (None)
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