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Theorem uniin2 28585
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 4514 . 2 𝑥𝐵 (𝐴𝑥) = (𝐴 𝑥𝐵 𝑥)
2 uniiun 4503 . . 3 𝐵 = 𝑥𝐵 𝑥
32ineq2i 3772 . 2 (𝐴 𝐵) = (𝐴 𝑥𝐵 𝑥)
41, 3eqtr4i 2634 1 𝑥𝐵 (𝐴𝑥) = (𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  cin 3538   cuni 4366   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 2033  ax-13 2233  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-uni 4367  df-iun 4451
This theorem is referenced by:  ldgenpisyslem1  29387
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