Theorem elunirab 3822

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 3821	. 2 |- ( A e. U. { x | ( x e. B /\ ph ) } <-> E. x ( A e. x /\ ( x e. B /\ ph ) ) )
2		df-rab 2464	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
3	2	unieqi 3819	. . 3 |- U. { x e. B | ph } = U. { x | ( x e. B /\ ph ) }
4	3	eleq2i 2244	. 2 |- ( A e. U. { x e. B | ph } <-> A e. U. { x | ( x e. B /\ ph ) } )
5		df-rex 2461	. . 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 1605	. . 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 1492   e. wcel 2148   {cab 2163   E.wrex 2456   {crab 2459   U.cuni 3809

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-rab 2464  df-v 2739  df-uni 3810

This theorem is referenced by: (None)