Theorem csbexa 4052

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
csbexa.1	|- A e. _V
csbexa.2	|- B e. _V

Assertion

Ref	Expression
csbexa	|- [_ A / x ]_ B e. _V

Proof of Theorem csbexa

Step	Hyp	Ref	Expression
1		csbexa.1	. . 3 |- A e. _V
2		csbexga 4051	. . 3 |- ( ( A e. _V /\ A. x B e. _V ) -> [_ A / x ]_ B e. _V )
3	1, 2	mpan 420	. 2 |- ( A. x B e. _V -> [_ A / x ]_ B e. _V )
4		csbexa.2	. 2 |- B e. _V
5	3, 4	mpg 1427	1 |- [_ A / x ]_ B e. _V

Syntax hints:   A.wal 1329   e. wcel 1480   _Vcvv 2681   [_csb 2998

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905  df-csb 2999

This theorem is referenced by:  dfmpo  6113