Theorem uniin2 32479

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	⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem uniin2

Step	Hyp	Ref	Expression
1		iunin2 5047	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥)
2		uniiun 5034	. . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
3	2	ineq2i 4192	. 2 ⊢ (𝐴 ∩ ∪ 𝐵) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥)
4	1, 3	eqtr4i 2761	1 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝐵)

Syntax hints:   = wceq 1540   ∩ cin 3925  ∪ cuni 4883  ∪ ciun 4967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rex 3061  df-rab 3416  df-v 3461  df-in 3933  df-uni 4884  df-iun 4969

This theorem is referenced by:  ssdifidllem  33417  ldgenpisyslem1  34140