Theorem csbexa 4213

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 4212	. . 3 |- ( ( A e. _V /\ A. x B e. _V ) -> [_ A / x ]_ B e. _V )
3	1, 2	mpan 424	. 2 |- ( A. x B e. _V -> [_ A / x ]_ B e. _V )
4		csbexa.2	. 2 |- B e. _V
5	3, 4	mpg 1497	1 |- [_ A / x ]_ B e. _V

Syntax hints:   A.wal 1393   e. wcel 2200   _Vcvv 2799   [_csb 3124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-csb 3125

This theorem is referenced by:  dfmpo  6375  rhmex  14129  fnpsr  14639  fnmpl  14665