Theorem elunirab 3749

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 3748	. 2 |- ( A e. U. { x | ( x e. B /\ ph ) } <-> E. x ( A e. x /\ ( x e. B /\ ph ) ) )
2		df-rab 2425	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
3	2	unieqi 3746	. . 3 |- U. { x e. B | ph } = U. { x | ( x e. B /\ ph ) }
4	3	eleq2i 2206	. 2 |- ( A e. U. { x e. B | ph } <-> A e. U. { x | ( x e. B /\ ph ) } )
5		df-rex 2422	. . 3 |- ( E. x e. B ( A e. x /\ ph ) <-> E. x ( x e. B /\ ( A e. x /\ ph ) ) )
6		an12 550	. . . 4 |- ( ( x e. B /\ ( A e. x /\ ph ) ) <-> ( A e. x /\ ( x e. B /\ ph ) ) )
7	6	exbii 1584	. . 3 |- ( E. x ( x e. B /\ ( A e. x /\ ph ) ) <-> E. x ( A e. x /\ ( x e. B /\ ph ) ) )
8	5, 7	bitri 183	. 2 |- ( E. x e. B ( A e. x /\ ph ) <-> E. x ( A e. x /\ ( x e. B /\ ph ) ) )
9	1, 4, 8	3bitr4i 211	1 |- ( A e. U. { x e. B | ph } <-> E. x e. B ( A e. x /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1468   e. wcel 1480   {cab 2125   E.wrex 2417   {crab 2420   U.cuni 3736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-rab 2425  df-v 2688  df-uni 3737

This theorem is referenced by: (None)