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Theorem uniin2 29671
Description: Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)
Assertion
Ref Expression
uniin2 𝑥𝐵 (𝐴𝑥) = (𝐴 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem uniin2
StepHypRef Expression
1 iunin2 4732 . 2 𝑥𝐵 (𝐴𝑥) = (𝐴 𝑥𝐵 𝑥)
2 uniiun 4721 . . 3 𝐵 = 𝑥𝐵 𝑥
32ineq2i 3950 . 2 (𝐴 𝐵) = (𝐴 𝑥𝐵 𝑥)
41, 3eqtr4i 2781 1 𝑥𝐵 (𝐴𝑥) = (𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1628  cin 3710   cuni 4584   ciun 4668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-v 3338  df-in 3718  df-uni 4585  df-iun 4670
This theorem is referenced by:  ldgenpisyslem1  30531
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