Theorem elunirab 3901

Description: Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)

Assertion

Ref	Expression
elunirab	|- ( A e. U. { x e. B | ph } <-> E. x e. B ( A e. x /\ ph ) )

Distinct variable group:   x, A

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

Proof of Theorem elunirab

Step	Hyp	Ref	Expression
1		eluniab 3900	. 2 |- ( A e. U. { x | ( x e. B /\ ph ) } <-> E. x ( A e. x /\ ( x e. B /\ ph ) ) )
2		df-rab 2517	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
3	2	unieqi 3898	. . 3 |- U. { x e. B | ph } = U. { x | ( x e. B /\ ph ) }
4	3	eleq2i 2296	. 2 |- ( A e. U. { x e. B | ph } <-> A e. U. { x | ( x e. B /\ ph ) } )
5		df-rex 2514	. . 3 |- ( E. x e. B ( A e. x /\ ph ) <-> E. x ( x e. B /\ ( A e. x /\ ph ) ) )
6		an12 561	. . . 4 |- ( ( x e. B /\ ( A e. x /\ ph ) ) <-> ( A e. x /\ ( x e. B /\ ph ) ) )
7	6	exbii 1651	. . 3 |- ( E. x ( x e. B /\ ( A e. x /\ ph ) ) <-> E. x ( A e. x /\ ( x e. B /\ ph ) ) )
8	5, 7	bitri 184	. 2 |- ( E. x e. B ( A e. x /\ ph ) <-> E. x ( A e. x /\ ( x e. B /\ ph ) ) )
9	1, 4, 8	3bitr4i 212	1 |- ( A e. U. { x e. B | ph } <-> E. x e. B ( A e. x /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1538   e. wcel 2200   {cab 2215   E.wrex 2509   {crab 2512   U.cuni 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-rab 2517  df-v 2801  df-uni 3889

This theorem is referenced by: (None)