Theorem csbexg 2008

Description: The existence of proper substitution into a class.

Assertion

Ref	Expression
csbexg	|- ((A e. C /\ A.x B e. D) -> [_A / x]_B e. V)

Proof of Theorem csbexg

Step	Hyp	Ref	Expression
1		a4sbc 1945	. . . . 5 |- (A e. C -> (A.x{y | y e. B} e. V -> [A / x]{y | y e. B} e. V))
2		elisset 1817	. . . . . . 7 |- (B e. D -> B e. V)
3		abid2 1580	. . . . . . 7 |- {y | y e. B} = B
4	2, 3	syl5eqel 1552	. . . . . 6 |- (B e. D -> {y | y e. B} e. V)
5	4	19.20i 992	. . . . 5 |- (A.x B e. D -> A.x{y | y e. B} e. V)
6	1, 5	syl5 21	. . . 4 |- (A e. C -> (A.x B e. D -> [A / x]{y | y e. B} e. V))
7	6	imp 350	. . 3 |- ((A e. C /\ A.x B e. D) -> [A / x]{y | y e. B} e. V)
8		ax-17 971	. . . . 5 |- (y e. V -> A.x y e. V)
9	8	sbcabel 1996	. . . 4 |- (A e. C -> ([A / x]{y | y e. B} e. V <-> {y | [A / x]y e. B} e. V))
10	9	adantr 389	. . 3 |- ((A e. C /\ A.x B e. D) -> ([A / x]{y | y e. B} e. V <-> {y | [A / x]y e. B} e. V))
11	7, 10	mpbid 195	. 2 |- ((A e. C /\ A.x B e. D) -> {y | [A / x]y e. B} e. V)
12		df-csb 2002	. 2 |- [_A / x]_B = {y | [A / x]y e. B}
13	11, 12	syl5eqel 1552	1 |- ((A e. C /\ A.x B e. D) -> [_A / x]_B e. V)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   e. wcel 958  [wsbc 1170  {cab 1463  Vcvv 1811  [_csb 2001

This theorem is referenced by:  csbex 2009  csbnestglem 2035  csbnestg 2036  csbnest1g 2037  sbcnestg 2038

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002

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