Theorem csbexg 1998

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 1935	. . . . 5 |- (A e. C -> (A.x{y | y e. B} e. V -> [A / x]{y | y e. B} e. V))
2		elisset 1808	. . . . . . 7 |- (B e. D -> B e. V)
3		abid2 1572	. . . . . . 7 |- {y | y e. B} = B
4	2, 3	syl5eqel 1544	. . . . . 6 |- (B e. D -> {y | y e. B} e. V)
5	4	19.20i 989	. . . . 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-17 968	. . . . 5 |- (y e. V -> A.x y e. V)
9	8	sbcabel 1986	. . . 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 1992	. 2 |- [_A / x]_B = {y | [A / x]y e. B}
13	11, 12	syl5eqel 1544	1 |- ((A e. C /\ A.x B e. D) -> [_A / x]_B e. V)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   e. wcel 955  [wsbc 1166  {cab 1456  Vcvv 1802  [_csb 1991

This theorem is referenced by:  csbex 1999  csbnestglem 2025  csbnestg 2026  csbnest1g 2027  sbcnestg 2028

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932  df-csb 1992

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