Theorem eluniab 3851

Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem eluniab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 3842	. 2 |- ( A e. U. { x | ph } <-> E. y ( A e. y /\ y e. { x | ph } ) )
2		nfv 1542	. . . 4 |- F/ x A e. y
3		nfsab1 2186	. . . 4 |- F/ x y e. { x | ph }
4	2, 3	nfan 1579	. . 3 |- F/ x ( A e. y /\ y e. { x | ph } )
5		nfv 1542	. . 3 |- F/ y ( A e. x /\ ph )
6		eleq2 2260	. . . 4 |- ( y = x -> ( A e. y <-> A e. x ) )
7		eleq1 2259	. . . . 5 |- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid 2184	. . . . 5 |- ( x e. { x | ph } <-> ph )
9	7, 8	bitrdi 196	. . . 4 |- ( y = x -> ( y e. { x | ph } <-> ph ) )
10	6, 9	anbi12d 473	. . 3 |- ( y = x -> ( ( A e. y /\ y e. { x | ph } ) <-> ( A e. x /\ ph ) ) )
11	4, 5, 10	cbvex 1770	. 2 |- ( E. y ( A e. y /\ y e. { x | ph } ) <-> E. x ( A e. x /\ ph ) )
12	1, 11	bitri 184	1 |- ( A e. U. { x | ph } <-> E. x ( A e. x /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1506   e. wcel 2167   {cab 2182   U.cuni 3839

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-uni 3840

This theorem is referenced by:  elunirab  3852  dfiun2g  3948  inuni  4188  snnex  4483  eliota  5246  elfv  5556  unielxp  6232  tfrlem9  6377  tfr0dm  6380  metrest  14742