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Theorem iunin1f 32341
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 5055 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) (Revised by Thierry Arnoux, 2-May-2020.)
Hypothesis
Ref Expression
iunin1f.1 𝑥𝐶
Assertion
Ref Expression
iunin1f 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)

Proof of Theorem iunin1f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iunin1f.1 . . . . . 6 𝑥𝐶
21nfcri 2885 . . . . 5 𝑥 𝑦𝐶
32r19.41 3255 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
4 elin 3960 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
54rexbii 3089 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
6 eliun 4995 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
76anbi1i 623 . . . 4 ((𝑦 𝑥𝐴 𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
83, 5, 73bitr4i 303 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦𝐶))
9 eliun 4995 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
10 elin 3960 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦𝐶))
118, 9, 103bitr4i 303 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ ( 𝑥𝐴 𝐵𝐶))
1211eqriv 2724 1 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wcel 2099  wnfc 2878  wrex 3065  cin 3943   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2164  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-rex 3066  df-v 3471  df-in 3951  df-iun 4993
This theorem is referenced by:  esum2dlem  33701
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