ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ab2rexex2 Unicode version

Theorem ab2rexex2 5998
Description: Existence of an existentially restricted class abstraction.  ph normally has free-variable parameters  x,  y, and  z. Compare abrexex2 5990. (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 5990 . 2  |-  { z  |  E. y  e.  B  ph }  e.  _V
51, 4abrexex2 5990 1  |-  { z  |  E. x  e.  A  E. y  e.  B  ph }  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1465   {cab 2103   E.wrex 2394   _Vcvv 2660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator