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Theorem iunin1f 29500
 Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4605 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 2787 . . . . 5 𝑥 𝑦𝐶
32r19.41 3119 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
4 elin 3829 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
54rexbii 3070 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
6 eliun 4556 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
76anbi1i 731 . . . 4 ((𝑦 𝑥𝐴 𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
83, 5, 73bitr4i 292 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦𝐶))
9 eliun 4556 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
10 elin 3829 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦𝐶))
118, 9, 103bitr4i 292 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ ( 𝑥𝐴 𝐵𝐶))
1211eqriv 2648 1 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Ⅎwnfc 2780  ∃wrex 2942   ∩ cin 3606  ∪ ciun 4552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-iun 4554 This theorem is referenced by:  esum2dlem  30282
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