Theorem ab2rexex 6292

Description: Existence of a class abstraction of existentially restricted sets. Variables x and y are normally free-variable parameters in the class expression substituted for C, which can be thought of as C ( x , y ). See comments for abrexex 6278. (Contributed by NM, 20-Sep-2011.)

Hypotheses

Ref	Expression
ab2rexex.1	|- A e. _V
ab2rexex.2	|- B e. _V

Assertion

Ref	Expression
ab2rexex	|- { z | E. x e. A E. y e. B z = C } e. _V

Distinct variable groups:   x, z, A   y, z, B   z, C

Allowed substitution hints:   A( y)   B( x)   C( x, y)

Proof of Theorem ab2rexex

Step	Hyp	Ref	Expression
1		ab2rexex.1	. 2 |- A e. _V
2		ab2rexex.2	. . 3 |- B e. _V
3	2	abrexex 6278	. 2 |- { z | E. y e. B z = C } e. _V
4	1, 3	abrexex2 6285	1 |- { z | E. x e. A E. y e. B z = C } e. _V

Syntax hints:   = wceq 1397   e. wcel 2202   {cab 2217   E.wrex 2511   _Vcvv 2802

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334

This theorem is referenced by:  fngsum  13470  igsumvalx  13471