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Theorem iunin1f 29922
 Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4793 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 2963 . . . . 5 𝑥 𝑦𝐶
32r19.41 3300 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
4 elin 4023 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
54rexbii 3251 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
6 eliun 4744 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
76anbi1i 619 . . . 4 ((𝑦 𝑥𝐴 𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
83, 5, 73bitr4i 295 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦𝐶))
9 eliun 4744 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
10 elin 4023 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦𝐶))
118, 9, 103bitr4i 295 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ ( 𝑥𝐴 𝐵𝐶))
1211eqriv 2822 1 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 386   = wceq 1658   ∈ wcel 2166  Ⅎwnfc 2956  ∃wrex 3118   ∩ cin 3797  ∪ ciun 4740 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-v 3416  df-in 3805  df-iun 4742 This theorem is referenced by:  esum2dlem  30699
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