Theorem csbexg 1979

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 1916	. . . . 5 |- (A e. C -> (A.x{y | y e. B} e. V -> [A / x]{y | y e. B} e. V))
2		elisset 1792	. . . . . . 7 |- (B e. D -> B e. V)
3		abid2 1556	. . . . . . 7 |- {y | y e. B} = B
4	2, 3	syl5eqel 1528	. . . . . 6 |- (B e. D -> {y | y e. B} e. V)
5	4	19.20i 968	. . . . 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 1190	. . . . 5 |- (y e. V -> A.x y e. V)
9	8	sbcabel 1967	. . . 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 1973	. 2 |- [_A / x]_B = {y | [A / x]y e. B}
13	11, 12	syl5eqel 1528	1 |- ((A e. C /\ A.x B e. D) -> [_A / x]_B e. V)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 950   e. wcel 1105  [wsbc 1153  {cab 1440  Vcvv 1786  [_csb 1972

This theorem is referenced by:  csbex 1980  csbnestglem 2006  csbnestg 2007  csbnest1g 2008  sbcnestg 2009

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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