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Theorem iunin1f 28563
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4503 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 nfcv 2750 . . . . . 6 𝑥𝑦
2 iunin1f.1 . . . . . 6 𝑥𝐶
31, 2nfel 2762 . . . . 5 𝑥 𝑦𝐶
43r19.41 3070 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
5 elin 3757 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
65rexbii 3022 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
7 eliun 4454 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
87anbi1i 726 . . . 4 ((𝑦 𝑥𝐴 𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
94, 6, 83bitr4i 290 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦𝐶))
10 eliun 4454 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
11 elin 3757 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦𝐶))
129, 10, 113bitr4i 290 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ ( 𝑥𝐴 𝐵𝐶))
1312eqriv 2606 1 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wcel 1976  wnfc 2737  wrex 2896  cin 3538   ciun 4449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-in 3546  df-iun 4451
This theorem is referenced by:  esum2dlem  29287
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