Theorem uniin2 28243

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	|- U_ x e. B ( A i^i x ) = ( A i^i U. B )

Distinct variable groups:   x, A   x, B

Proof of Theorem uniin2

Step	Hyp	Ref	Expression
1		iunin2 4333	. 2 |- U_ x e. B ( A i^i x ) = ( A i^i U_ x e. B x )
2		uniiun 4322	. . 3 |- U. B = U_ x e. B x
3	2	ineq2i 3622	. 2 |- ( A i^i U. B ) = ( A i^i U_ x e. B x )
4	1, 3	eqtr4i 2496	1 |- U_ x e. B ( A i^i x ) = ( A i^i U. B )

Syntax hints:   = wceq 1452   i^i cin 3389   U.cuni 4190   U_ciun 4269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-v 3033  df-in 3397  df-uni 4191  df-iun 4271

This theorem is referenced by:  ldgenpisyslem1  29059