Theorem abrexex2 6209

Description: Existence of an existentially restricted class abstraction. ph is normally has free-variable parameters x and y. See also abrexex 6202. (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 1551	. . . 4 |- F/ z E. x e. A ph
2		nfcv 2348	. . . . 5 |- F/_ y A
3		nfs1v 1967	. . . . 5 |- F/ y [ z / y ] ph
4	2, 3	nfrexw 2545	. . . 4 |- F/ y E. x e. A [ z / y ] ph
5		sbequ12 1794	. . . . 5 |- ( y = z -> ( ph <-> [ z / y ] ph ) )
6	5	rexbidv 2507	. . . 4 |- ( y = z -> ( E. x e. A ph <-> E. x e. A [ z / y ] ph ) )
7	1, 4, 6	cbvab 2329	. . 3 |- { y | E. x e. A ph } = { z | E. x e. A [ z / y ] ph }
8		df-clab 2192	. . . . 5 |- ( z e. { y | ph } <-> [ z / y ] ph )
9	8	rexbii 2513	. . . 4 |- ( E. x e. A z e. { y | ph } <-> E. x e. A [ z / y ] ph )
10	9	abbii 2321	. . 3 |- { z | E. x e. A z e. { y | ph } } = { z | E. x e. A [ z / y ] ph }
11	7, 10	eqtr4i 2229	. 2 |- { y | E. x e. A ph } = { z | E. x e. A z e. { y | ph } }
12		df-iun 3929	. . 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 6208	. . 3 |- U_ x e. A { y | ph } e. _V
16	12, 15	eqeltrri 2279	. 2 |- { z | E. x e. A z e. { y | ph } } e. _V
17	11, 16	eqeltri 2278	1 |- { y | E. x e. A ph } e. _V

Syntax hints:   [wsb 1785   e. wcel 2176   {cab 2191   E.wrex 2485   _Vcvv 2772   U_ciun 3927

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279

This theorem is referenced by:  abexssex  6210  abexex  6211  oprabrexex2  6215  ab2rexex  6216  ab2rexex2  6217