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Theorem ab2rexex 6772
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 6755. (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 6755 . 2  |-  { z  |  E. y  e.  B  z  =  C }  e.  _V
41, 3abrexex2 6762 1  |-  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815   _Vcvv 3113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594
This theorem is referenced by:  plyval  22325  pstmfval  27511  pstmxmet  27512
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