Theorem ab2rexex 6216

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 6202. (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 6202	. 2 |- { z | E. y e. B z = C } e. _V
4	1, 3	abrexex2 6209	1 |- { z | E. x e. A E. y e. B z = C } e. _V

Syntax hints:   = wceq 1373   e. wcel 2176   {cab 2191   E.wrex 2485   _Vcvv 2772

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:  fngsum  13220  igsumvalx  13221