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Theorem ab2rexex 6126
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 5883. (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 5883 . 2  |-  { z  |  E. y  e.  B  z  =  C }  e.  _V
41, 3abrexex2 5901 1  |-  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1647    e. wcel 1715   {cab 2352   E.wrex 2629   _Vcvv 2873
This theorem is referenced by:  plyval  19790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366
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