Theorem sbcel2gv 1991

Description: Class substitution into a membership relation.

Assertion

Ref	Expression
sbcel2gv	|- (B e. C -> ([B / x]A e. x <-> A e. B))

Distinct variable group:   x,A

Proof of Theorem sbcel2gv

Step	Hyp	Ref	Expression
1		ax-17 975	. . . 4 |- (A e. y -> A.x A e. y)
2		eleq2 1542	. . . 4 |- (x = y -> (A e. x <-> A e. y))
3	1, 2	sbie 1200	. . 3 |- ([y / x]A e. x <-> A e. y)
4	3	sbcbii 1988	. 2 |- (B e. C -> ([B / y][y / x]A e. x <-> [B / y]A e. y))
5		sbccog 1959	. 2 |- (B e. C -> ([B / y][y / x]A e. x <-> [B / x]A e. x))
6		elisset 1824	. . 3 |- (B e. C -> B e. V)
7		elex 1826	. . . 4 |- (B e. V -> E.y y = B)
8		ax-17 975	. . . . . . . 8 |- (z e. B -> A.y z e. B)
9	8	hbsbc1 1956	. . . . . . 7 |- ((B e. V -> [B / y]A e. y) -> A.y(B e. V -> [B / y]A e. y))
10		ax-17 975	. . . . . . 7 |- ((B e. V -> A e. B) -> A.y(B e. V -> A e. B))
11	9, 10	hbbi 1014	. . . . . 6 |- (((B e. V -> [B / y]A e. y) <-> (B e. V -> A e. B)) -> A.y((B e. V -> [B / y]A e. y) <-> (B e. V -> A e. B)))
12		sbceq1a 1951	. . . . . . . 8 |- (y = B -> (A e. y <-> [B / y]A e. y))
13		eleq2 1542	. . . . . . . 8 |- (y = B -> (A e. y <-> A e. B))
14	12, 13	bitr3d 533	. . . . . . 7 |- (y = B -> ([B / y]A e. y <-> A e. B))
15	14	imbi2d 615	. . . . . 6 |- (y = B -> ((B e. V -> [B / y]A e. y) <-> (B e. V -> A e. B)))
16	11, 15	19.23ai 1068	. . . . 5 |- (E.y y = B -> ((B e. V -> [B / y]A e. y) <-> (B e. V -> A e. B)))
17	16	pm5.74rd 591	. . . 4 |- (E.y y = B -> (B e. V -> ([B / y]A e. y <-> A e. B)))
18	7, 17	mpcom 49	. . 3 |- (B e. V -> ([B / y]A e. y <-> A e. B))
19	6, 18	syl 10	. 2 |- (B e. C -> ([B / y]A e. y <-> A e. B))
20	4, 5, 19	3bitr3d 551	1 |- (B e. C -> ([B / x]A e. x <-> A e. B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 960   e. wcel 962  E.wex 984  [wsbc 1174  Vcvv 1818

This theorem is referenced by:  csbvarg 2032

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-v 1819  df-sbc 1949

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