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

Theorem csbex 3254
Description: The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypotheses
Ref Expression
csbex.1  |-  A  e. 
_V
csbex.2  |-  B  e. 
_V
Assertion
Ref Expression
csbex  |-  [_ A  /  x ]_ B  e. 
_V

Proof of Theorem csbex
StepHypRef Expression
1 csbex.1 . . 3  |-  A  e. 
_V
2 csbexg 3253 . . 3  |-  ( ( A  e.  _V  /\  A. x  B  e.  _V )  ->  [_ A  /  x ]_ B  e.  _V )
31, 2mpan 652 . 2  |-  ( A. x  B  e.  _V  ->  [_ A  /  x ]_ B  e.  _V )
4 csbex.2 . 2  |-  B  e. 
_V
53, 4mpg 1557 1  |-  [_ A  /  x ]_ B  e. 
_V
Colors of variables: wff set class
Syntax hints:   A.wal 1549    e. wcel 1725   _Vcvv 2948   [_csb 3243
This theorem is referenced by:  dfmpt2  6429  cantnfdm  7611  cantnff  7621  ruclem1  12822  pcmpt  13253  cidffn  13895  natffn  14138  fnxpc  14265  evlfcl  14311  odf  15167  itgfsum  19710  vmaf  20894  bpolylem  26086  aomclem6  27125
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154  df-csb 3244
  Copyright terms: Public domain W3C validator