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Theorem uniin2 32473
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 5071 . 2 𝑥𝐵 (𝐴𝑥) = (𝐴 𝑥𝐵 𝑥)
2 uniiun 5058 . . 3 𝐵 = 𝑥𝐵 𝑥
32ineq2i 4207 . 2 (𝐴 𝐵) = (𝐴 𝑥𝐵 𝑥)
41, 3eqtr4i 2757 1 𝑥𝐵 (𝐴𝑥) = (𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  cin 3945   cuni 4905   ciun 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rex 3061  df-rab 3420  df-v 3464  df-in 3953  df-uni 4906  df-iun 4995
This theorem is referenced by:  ssdifidllem  33337  ldgenpisyslem1  34009
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