Theorem eluni2f 39286

Description: Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
eluni2f.1	|- F/_ x A
eluni2f.2	|- F/_ x B

Assertion

Ref	Expression
eluni2f	|- ( A e. U. B <-> E. x e. B A e. x )

Distinct variable group:   A, B

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem eluni2f

Step	Hyp	Ref	Expression
1		exancom 1787	. 2 |- ( E. x ( A e. x /\ x e. B ) <-> E. x ( x e. B /\ A e. x ) )
2		eluni2f.1	. . 3 |- F/_ x A
3		eluni2f.2	. . 3 |- F/_ x B
4	2, 3	elunif 39175	. 2 |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )
5		df-rex 2918	. 2 |- ( E. x e. B A e. x <-> E. x ( x e. B /\ A e. x ) )
6	1, 4, 5	3bitr4i 292	1 |- ( A e. U. B <-> E. x e. B A e. x )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   F/_wnfc 2751   E.wrex 2913   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-uni 4437

This theorem is referenced by:  smfresal  40995  smfpimbor1lem2  41006