Theorem csbex 2005

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 961	. 2 |- A.x B e. V
4		csbexg 2004	. 2 |- ((A e. V /\ A.x B e. V) -> [_A / x]_B e. V)
5	1, 3, 4	mp2an 696	1 |- [_A / x]_B e. V

Syntax hints:  A.wal 952   e. wcel 956  Vcvv 1807  [_csb 1997

This theorem is referenced by:  fvopab4sf 3773  fvopabs 3783  fopabcos 3824  fsum1slem 6954  fsump1f 6957  fsump1slem 6958  csbfsumlem 6972

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938  df-csb 1998

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