Theorem abrexex2 6232

Description: Existence of an existentially restricted class abstraction. ph is normally has free-variable parameters x and y. See also abrexex 6225. (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 1552	. . . 4 |- F/ z E. x e. A ph
2		nfcv 2350	. . . . 5 |- F/_ y A
3		nfs1v 1968	. . . . 5 |- F/ y [ z / y ] ph
4	2, 3	nfrexw 2547	. . . 4 |- F/ y E. x e. A [ z / y ] ph
5		sbequ12 1795	. . . . 5 |- ( y = z -> ( ph <-> [ z / y ] ph ) )
6	5	rexbidv 2509	. . . 4 |- ( y = z -> ( E. x e. A ph <-> E. x e. A [ z / y ] ph ) )
7	1, 4, 6	cbvab 2331	. . 3 |- { y | E. x e. A ph } = { z | E. x e. A [ z / y ] ph }
8		df-clab 2194	. . . . 5 |- ( z e. { y | ph } <-> [ z / y ] ph )
9	8	rexbii 2515	. . . 4 |- ( E. x e. A z e. { y | ph } <-> E. x e. A [ z / y ] ph )
10	9	abbii 2323	. . 3 |- { z | E. x e. A z e. { y | ph } } = { z | E. x e. A [ z / y ] ph }
11	7, 10	eqtr4i 2231	. 2 |- { y | E. x e. A ph } = { z | E. x e. A z e. { y | ph } }
12		df-iun 3943	. . 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 6231	. . 3 |- U_ x e. A { y | ph } e. _V
16	12, 15	eqeltrri 2281	. 2 |- { z | E. x e. A z e. { y | ph } } e. _V
17	11, 16	eqeltri 2280	1 |- { y | E. x e. A ph } e. _V

Syntax hints:   [wsb 1786   e. wcel 2178   {cab 2193   E.wrex 2487   _Vcvv 2776   U_ciun 3941

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298

This theorem is referenced by:  abexssex  6233  abexex  6234  oprabrexex2  6238  ab2rexex  6239  ab2rexex2  6240