Theorem csbex 2012

Description: The existence of proper substitution into a class.

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	. 2 |- A e. V
2		csbex.2	. . 3 |- B e. V
3	2	ax-gen 965	. 2 |- A.x B e. V
4		csbexg 2011	. 2 |- ((A e. V /\ A.x B e. V) -> [_A / x]_B e. V)
5	1, 3, 4	mp2an 699	1 |- [_A / x]_B e. V

Syntax hints:  A.wal 956   e. wcel 960  Vcvv 1814  [_csb 2004

This theorem is referenced by:  fvopab4sf 3788  fvopabs 3798  fopabcos 3839  fsum1slem 7008  fsump1f 7011  fsump1slem 7012  csbfsumlem 7026

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-sbc 1945  df-csb 2005

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