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Theorem uniin2 30311
 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 4968 . 2 𝑥𝐵 (𝐴𝑥) = (𝐴 𝑥𝐵 𝑥)
2 uniiun 4957 . . 3 𝐵 = 𝑥𝐵 𝑥
32ineq2i 4160 . 2 (𝐴 𝐵) = (𝐴 𝑥𝐵 𝑥)
41, 3eqtr4i 2848 1 𝑥𝐵 (𝐴𝑥) = (𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∩ cin 3907  ∪ cuni 4813  ∪ ciun 4894 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-in 3915  df-uni 4814  df-iun 4896 This theorem is referenced by:  ldgenpisyslem1  31496
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