Theorem abrexex2 6267

Description: Existence of an existentially restricted class abstraction. ph is normally has free-variable parameters x and y. See also abrexex 6260. (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 1574	. . . 4 |- F/ z E. x e. A ph
2		nfcv 2372	. . . . 5 |- F/_ y A
3		nfs1v 1990	. . . . 5 |- F/ y [ z / y ] ph
4	2, 3	nfrexw 2569	. . . 4 |- F/ y E. x e. A [ z / y ] ph
5		sbequ12 1817	. . . . 5 |- ( y = z -> ( ph <-> [ z / y ] ph ) )
6	5	rexbidv 2531	. . . 4 |- ( y = z -> ( E. x e. A ph <-> E. x e. A [ z / y ] ph ) )
7	1, 4, 6	cbvab 2353	. . 3 |- { y | E. x e. A ph } = { z | E. x e. A [ z / y ] ph }
8		df-clab 2216	. . . . 5 |- ( z e. { y | ph } <-> [ z / y ] ph )
9	8	rexbii 2537	. . . 4 |- ( E. x e. A z e. { y | ph } <-> E. x e. A [ z / y ] ph )
10	9	abbii 2345	. . 3 |- { z | E. x e. A z e. { y | ph } } = { z | E. x e. A [ z / y ] ph }
11	7, 10	eqtr4i 2253	. 2 |- { y | E. x e. A ph } = { z | E. x e. A z e. { y | ph } }
12		df-iun 3966	. . 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 6266	. . 3 |- U_ x e. A { y | ph } e. _V
16	12, 15	eqeltrri 2303	. 2 |- { z | E. x e. A z e. { y | ph } } e. _V
17	11, 16	eqeltri 2302	1 |- { y | E. x e. A ph } e. _V

Syntax hints:   [wsb 1808   e. wcel 2200   {cab 2215   E.wrex 2509   _Vcvv 2799   U_ciun 3964

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325

This theorem is referenced by:  abexssex  6268  abexex  6269  oprabrexex2  6273  ab2rexex  6274  ab2rexex2  6275