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Theorem ab2rexex 6189
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 5946. (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 5946 . 2  |-  { z  |  E. y  e.  B  z  =  C }  e.  _V
41, 3abrexex2 5964 1  |-  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   {cab 2394   E.wrex 2671   _Vcvv 2920
This theorem is referenced by:  plyval  20069  pstmfval  24248  pstmxmet  24249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425
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