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Theorem ab2rexex2 6219
Description: Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x,  y, and  z. Compare abrexex2 5993. (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 5993 . 2  |-  { z  |  E. y  e.  B  ph }  e.  _V
51, 4abrexex2 5993 1  |-  { z  |  E. x  e.  A  E. y  e.  B  ph }  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   {cab 2421   E.wrex 2698   _Vcvv 2948
This theorem is referenced by:  brdom7disj  8399  brdom6disj  8400  lineset  30436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454
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