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Theorem iunin1f 32577
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 5062 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 2894 . . . . 5 𝑥 𝑦𝐶
32r19.41 3260 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
4 elin 3978 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
54rexbii 3091 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
6 eliun 4999 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
76anbi1i 624 . . . 4 ((𝑦 𝑥𝐴 𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
83, 5, 73bitr4i 303 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦𝐶))
9 eliun 4999 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
10 elin 3978 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦𝐶))
118, 9, 103bitr4i 303 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ ( 𝑥𝐴 𝐵𝐶))
1211eqriv 2731 1 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1536  wcel 2105  wnfc 2887  wrex 3067  cin 3961   ciun 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-rex 3068  df-v 3479  df-in 3969  df-iun 4997
This theorem is referenced by:  esum2dlem  34072
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