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Theorem ab2rexex2 6168
Description: Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x,  y, and  z. Compare abrexex2 5942. (Contributed by NM, 20-Sep-2011.)
Hypotheses
Ref Expression
ab2rexex2.1  |-  A  e. 
_V
ab2rexex2.2  |-  B  e. 
_V
ab2rexex2.3  |-  { z  |  ph }  e.  _V
Assertion
Ref Expression
ab2rexex2  |-  { z  |  E. x  e.  A  E. y  e.  B  ph }  e.  _V
Distinct variable groups:    x, z, A    y, z, B
Allowed substitution hints:    ph( x, y, z)    A( y)    B( x)

Proof of Theorem ab2rexex2
StepHypRef Expression
1 ab2rexex2.1 . 2  |-  A  e. 
_V
2 ab2rexex2.2 . . 3  |-  B  e. 
_V
3 ab2rexex2.3 . . 3  |-  { z  |  ph }  e.  _V
42, 3abrexex2 5942 . 2  |-  { z  |  E. y  e.  B  ph }  e.  _V
51, 4abrexex2 5942 1  |-  { z  |  E. x  e.  A  E. y  e.  B  ph }  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1717   {cab 2375   E.wrex 2652   _Vcvv 2901
This theorem is referenced by:  brdom7disj  8344  brdom6disj  8345  lineset  29854
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404
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