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Theorem ab2rexex2 6777
Description: Existence of an existentially restricted class abstraction.  ph normally has free-variable parameters  x,  y, and  z. Compare abrexex2 6766. (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 6766 . 2  |-  { z  |  E. y  e.  B  ph }  e.  _V
51, 4abrexex2 6766 1  |-  { z  |  E. x  e.  A  E. y  e.  B  ph }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1804   {cab 2428   E.wrex 2794   _Vcvv 3095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586
This theorem is referenced by:  brdom7disj  8912  brdom6disj  8913  lineset  35202
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