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Theorem ab2rexex2 6671
Description: Existence of an existentially restricted class abstraction.  ph normally has free-variable parameters  x,  y, and  z. Compare abrexex2 6660. (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 6660 . 2  |-  { z  |  E. y  e.  B  ph }  e.  _V
51, 4abrexex2 6660 1  |-  { z  |  E. x  e.  A  E. y  e.  B  ph }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758   {cab 2436   E.wrex 2796   _Vcvv 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526
This theorem is referenced by:  brdom7disj  8801  brdom6disj  8802  lineset  33690
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