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Theorem uniin2 29916
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 4804 . 2 𝑥𝐵 (𝐴𝑥) = (𝐴 𝑥𝐵 𝑥)
2 uniiun 4793 . . 3 𝐵 = 𝑥𝐵 𝑥
32ineq2i 4038 . 2 (𝐴 𝐵) = (𝐴 𝑥𝐵 𝑥)
41, 3eqtr4i 2852 1 𝑥𝐵 (𝐴𝑥) = (𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1658  cin 3797   cuni 4658   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-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-uni 4659  df-iun 4742
This theorem is referenced by:  ldgenpisyslem1  30771
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