MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ab2rexex Structured version   Unicode version

Theorem ab2rexex 6774
Description: Existence of a class abstraction of existentially restricted sets. Variables  x and  y are normally free-variable parameters in the class expression substituted for  C, which can be thought of as  C ( x ,  y ). See comments for abrexex 6757. (Contributed by NM, 20-Sep-2011.)
Hypotheses
Ref Expression
ab2rexex.1  |-  A  e. 
_V
ab2rexex.2  |-  B  e. 
_V
Assertion
Ref Expression
ab2rexex  |-  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  e.  _V
Distinct variable groups:    x, z, A    y, z, B    z, C
Allowed substitution hints:    A( y)    B( x)    C( x, y)

Proof of Theorem ab2rexex
StepHypRef Expression
1 ab2rexex.1 . 2  |-  A  e. 
_V
2 ab2rexex.2 . . 3  |-  B  e. 
_V
32abrexex 6757 . 2  |-  { z  |  E. y  e.  B  z  =  C }  e.  _V
41, 3abrexex2 6764 1  |-  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    e. wcel 1842   {cab 2387   E.wrex 2754   _Vcvv 3058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576
This theorem is referenced by:  plyval  22880  pstmfval  28314  pstmxmet  28315
  Copyright terms: Public domain W3C validator