Theorem csbexa 4111

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 4110	. . 3 |- ( ( A e. _V /\ A. x B e. _V ) -> [_ A / x ]_ B e. _V )
3	1, 2	mpan 421	. 2 |- ( A. x B e. _V -> [_ A / x ]_ B e. _V )
4		csbexa.2	. 2 |- B e. _V
5	3, 4	mpg 1439	1 |- [_ A / x ]_ B e. _V

Syntax hints:   A.wal 1341   e. wcel 2136   _Vcvv 2726   [_csb 3045

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by:  dfmpo  6191