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Theorem ab2rexex 6229
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 5986. (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
StepHypRef Expression
1 ab2rexex.1 . 2  |-  A  e. 
_V
2 ab2rexex.2 . . 3  |-  B  e. 
_V
32abrexex 5986 . 2  |-  { z  |  E. y  e.  B  z  =  C }  e.  _V
41, 3abrexex2 6004 1  |-  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   {cab 2424   E.wrex 2708   _Vcvv 2958
This theorem is referenced by:  plyval  20117  pstmfval  24296  pstmxmet  24297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465
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