Theorem abrexex2 6092

Description: Existence of an existentially restricted class abstraction. ph is normally has free-variable parameters x and y. See also abrexex 6085. (Contributed by NM, 12-Sep-2004.)

Hypotheses

Ref	Expression
abrexex2.1	|- A e. _V
abrexex2.2	|- { y | ph } e. _V

Assertion

Ref	Expression
abrexex2	|- { y | E. x e. A ph } e. _V

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem abrexex2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1516	. . . 4 |- F/ z E. x e. A ph
2		nfcv 2308	. . . . 5 |- F/_ y A
3		nfs1v 1927	. . . . 5 |- F/ y [ z / y ] ph
4	2, 3	nfrexxy 2505	. . . 4 |- F/ y E. x e. A [ z / y ] ph
5		sbequ12 1759	. . . . 5 |- ( y = z -> ( ph <-> [ z / y ] ph ) )
6	5	rexbidv 2467	. . . 4 |- ( y = z -> ( E. x e. A ph <-> E. x e. A [ z / y ] ph ) )
7	1, 4, 6	cbvab 2290	. . 3 |- { y | E. x e. A ph } = { z | E. x e. A [ z / y ] ph }
8		df-clab 2152	. . . . 5 |- ( z e. { y | ph } <-> [ z / y ] ph )
9	8	rexbii 2473	. . . 4 |- ( E. x e. A z e. { y | ph } <-> E. x e. A [ z / y ] ph )
10	9	abbii 2282	. . 3 |- { z | E. x e. A z e. { y | ph } } = { z | E. x e. A [ z / y ] ph }
11	7, 10	eqtr4i 2189	. 2 |- { y | E. x e. A ph } = { z | E. x e. A z e. { y | ph } }
12		df-iun 3868	. . 3 |- U_ x e. A { y | ph } = { z | E. x e. A z e. { y | ph } }
13		abrexex2.1	. . . 4 |- A e. _V
14		abrexex2.2	. . . 4 |- { y | ph } e. _V
15	13, 14	iunex 6091	. . 3 |- U_ x e. A { y | ph } e. _V
16	12, 15	eqeltrri 2240	. 2 |- { z | E. x e. A z e. { y | ph } } e. _V
17	11, 16	eqeltri 2239	1 |- { y | E. x e. A ph } e. _V

Syntax hints:   [wsb 1750   e. wcel 2136   {cab 2151   E.wrex 2445   _Vcvv 2726   U_ciun 3866

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by:  abexssex  6093  abexex  6094  oprabrexex2  6098  ab2rexex  6099  ab2rexex2  6100