Theorem sbcel2gv 1977

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-17 969	. . . 4 |- (A e. y -> A.x A e. y)
2		eleq2 1532	. . . 4 |- (x = y -> (A e. x <-> A e. y))
3	1, 2	sbie 1194	. . 3 |- ([y / x]A e. x <-> A e. y)
4	3	sbcbii 1974	. 2 |- (B e. C -> ([B / y][y / x]A e. x <-> [B / y]A e. y))
5		sbccog 1948	. 2 |- (B e. C -> ([B / y][y / x]A e. x <-> [B / x]A e. x))
6		elisset 1813	. . 3 |- (B e. C -> B e. V)
7		elex 1815	. . . 4 |- (B e. V -> E.y y = B)
8		ax-17 969	. . . . . . . 8 |- (z e. B -> A.y z e. B)
9	8	hbsbc1 1945	. . . . . . 7 |- ((B e. V -> [B / y]A e. y) -> A.y(B e. V -> [B / y]A e. y))
10		ax-17 969	. . . . . . 7 |- ((B e. V -> A e. B) -> A.y(B e. V -> A e. B))
11	9, 10	hbbi 1008	. . . . . 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 1940	. . . . . . . 8 |- (y = B -> (A e. y <-> [B / y]A e. y))
13		eleq2 1532	. . . . . . . 8 |- (y = B -> (A e. y <-> A e. B))
14	12, 13	bitr3d 529	. . . . . . 7 |- (y = B -> ([B / y]A e. y <-> A e. B))
15	14	imbi2d 611	. . . . . 6 |- (y = B -> ((B e. V -> [B / y]A e. y) <-> (B e. V -> A e. B)))
16	11, 15	19.23ai 1062	. . . . 5 |- (E.y y = B -> ((B e. V -> [B / y]A e. y) <-> (B e. V -> A e. B)))
17	16	pm5.74rd 587	. . . 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 547	1 |- (B e. C -> ([B / x]A e. x <-> A e. B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 954   e. wcel 956  E.wex 978  [wsbc 1168  Vcvv 1807

This theorem is referenced by:  csbvarg 2017

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938

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