Theorem csbex 1980

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 955	. 2 |- A.x B e. V
4		csbexg 1979	. 2 |- ((A e. V /\ A.x B e. V) -> [_A / x]_B e. V)
5	1, 3, 4	mp2an 694	1 |- [_A / x]_B e. V

Syntax hints:  A.wal 950   e. wcel 1105  Vcvv 1786  [_csb 1972

This theorem is referenced by:  fvopab4sf 3721  fvopabs 3731  fopabcos 3772  fsum1slem 6897  fsump1f 6900  fsump1slem 6901  csbfsumlem 6915

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-sbc 1913  df-csb 1973

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