Theorem sbcel2gv 1981

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-17 971	. . . 4 |- (A e. y -> A.x A e. y)
2		eleq2 1535	. . . 4 |- (x = y -> (A e. x <-> A e. y))
3	1, 2	sbie 1196	. . 3 |- ([y / x]A e. x <-> A e. y)
4	3	sbcbii 1978	. 2 |- (B e. C -> ([B / y][y / x]A e. x <-> [B / y]A e. y))
5		sbccog 1952	. 2 |- (B e. C -> ([B / y][y / x]A e. x <-> [B / x]A e. x))
6		elisset 1817	. . 3 |- (B e. C -> B e. V)
7		elex 1819	. . . 4 |- (B e. V -> E.y y = B)
8		ax-17 971	. . . . . . . 8 |- (z e. B -> A.y z e. B)
9	8	hbsbc1 1949	. . . . . . 7 |- ((B e. V -> [B / y]A e. y) -> A.y(B e. V -> [B / y]A e. y))
10		ax-17 971	. . . . . . 7 |- ((B e. V -> A e. B) -> A.y(B e. V -> A e. B))
11	9, 10	hbbi 1010	. . . . . 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 1944	. . . . . . . 8 |- (y = B -> (A e. y <-> [B / y]A e. y))
13		eleq2 1535	. . . . . . . 8 |- (y = B -> (A e. y <-> A e. B))
14	12, 13	bitr3d 530	. . . . . . 7 |- (y = B -> ([B / y]A e. y <-> A e. B))
15	14	imbi2d 612	. . . . . 6 |- (y = B -> ((B e. V -> [B / y]A e. y) <-> (B e. V -> A e. B)))
16	11, 15	19.23ai 1064	. . . . 5 |- (E.y y = B -> ((B e. V -> [B / y]A e. y) <-> (B e. V -> A e. B)))
17	16	pm5.74rd 588	. . . 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 548	1 |- (B e. C -> ([B / x]A e. x <-> A e. B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  E.wex 980  [wsbc 1170  Vcvv 1811

This theorem is referenced by:  csbvarg 2021

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

Copyright terms: Public domain