Theorem csbex 2549

Description: The existence of proper substitution into a class. (The proof was 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

Step	Hyp	Ref	Expression
1		csbex.1	. . 3 |- A e. _V
2		csbexg 2548	. . 3 |- ((A e. _V /\ A.x B e. _V) -> [_A / x]_B e. _V)
3	1, 2	mpan 759	. 2 |- (A.x B e. _V -> [_A / x]_B e. _V)
4		csbex.2	. 2 |- B e. _V
5	3, 4	mpg 1332	1 |- [_A / x]_B e. _V

Syntax hints:  A.wal 1296   e. wcel 1300  _Vcvv 2292  [_csb 2540

This theorem is referenced by:  eufromeq4 3831  fvopab4sf 4745  fvopabs 4755  fopabcos 4806  iunfoprab 5072  fsum1slem 8268  fsump1fi 8271  fsump1slem 8272  csbfsumlem 8286  oprabco 10159  fprod1slem 14676  fprodp1fi 14680  fprodp1slem 14681  oprpiece1res1 15880  oprpiece1res2 15881

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454  df-csb 2541

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