Theorem sbcel2 1467

Description: Class substitution into a membership relation.

Assertion

Ref	Expression
sbcel2	|- (B e. C -> ([B / x]A e. x <-> A e. B))

Distinct variable group(s):   x,A   x,B

Proof of Theorem sbcel2

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 |- (B e. C -> B e. V)
2		elex 1356	. . 3 |- (B e. V -> E.x x = B)
3		ax-17 925	. . . . . . 7 |- (y e. B -> A.x y e. B)
4	3	hbsbc 1446	. . . . . 6 |- ((B e. V -> [B / x]A e. x) -> A.x(B e. V -> [B / x]A e. x))
5		ax-17 925	. . . . . 6 |- ((B e. V -> A e. B) -> A.x(B e. V -> A e. B))
6	4, 5	hbbi 705	. . . . 5 |- (((B e. V -> [B / x]A e. x) <-> (B e. V -> A e. B)) -> A.x((B e. V -> [B / x]A e. x) <-> (B e. V -> A e. B)))
7		sbceq1 1443	. . . . . . 7 |- (x = B -> (A e. x <-> [B / x]A e. x))
8		eleq2 1150	. . . . . . 7 |- (x = B -> (A e. x <-> A e. B))
9	7, 8	bitr3d 408	. . . . . 6 |- (x = B -> ([B / x]A e. x <-> A e. B))
10	9	imbi2d 464	. . . . 5 |- (x = B -> ((B e. V -> [B / x]A e. x) <-> (B e. V -> A e. B)))
11	6, 10	19.23ai 746	. . . 4 |- (E.x x = B -> ((B e. V -> [B / x]A e. x) <-> (B e. V -> A e. B)))
12	11	pm5.74rd 446	. . 3 |- (E.x x = B -> (B e. V -> ([B / x]A e. x <-> A e. B)))
13	2, 12	mpcom 49	. 2 |- (B e. V -> ([B / x]A e. x <-> A e. B))
14	1, 13	syl 12	1 |- (B e. C -> ([B / x]A e. x <-> A e. B))

Syntax hints:   -> wi 2   <-> wb 127  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348  [wsbc 1440

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-sbc 1441

metamath.org