Theorem ab2rexex 6364

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 6346. (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 6346	. 2 |- { z | E. y e. B z = C } e. _V
4	1, 3	abrexex2 6353	1 |- { z | E. x e. A E. y e. B z = C } e. _V

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209   {cab 2224   E.wrex 2529   _Vcvv 2821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385

This theorem is used by:  fngzsum  13708  gzsumvalx  13709