Theorem elunirab 3593

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 3592	. 2 |- ( A e. U. { x | ( x e. B /\ ph ) } <-> E. x ( A e. x /\ ( x e. B /\ ph ) ) )
2		df-rab 2315	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
3	2	unieqi 3590	. . 3 |- U. { x e. B | ph } = U. { x | ( x e. B /\ ph ) }
4	3	eleq2i 2104	. 2 |- ( A e. U. { x e. B | ph } <-> A e. U. { x | ( x e. B /\ ph ) } )
5		df-rex 2312	. . 3 |- ( E. x e. B ( A e. x /\ ph ) <-> E. x ( x e. B /\ ( A e. x /\ ph ) ) )
6		an12 495	. . . 4 |- ( ( x e. B /\ ( A e. x /\ ph ) ) <-> ( A e. x /\ ( x e. B /\ ph ) ) )
7	6	exbii 1496	. . 3 |- ( E. x ( x e. B /\ ( A e. x /\ ph ) ) <-> E. x ( A e. x /\ ( x e. B /\ ph ) ) )
8	5, 7	bitri 173	. 2 |- ( E. x e. B ( A e. x /\ ph ) <-> E. x ( A e. x /\ ( x e. B /\ ph ) ) )
9	1, 4, 8	3bitr4i 201	1 |- ( A e. U. { x e. B | ph } <-> E. x e. B ( A e. x /\ ph ) )

Syntax hints:   /\ wa 97   <-> wb 98   E.wex 1381   e. wcel 1393   {cab 2026   E.wrex 2307   {crab 2310   U.cuni 3580

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-rab 2315  df-v 2559  df-uni 3581

This theorem is referenced by: (None)