Theorem sbcel1gv 1976

Description: Class substitution into a membership relation.

Assertion

Ref	Expression
sbcel1gv	|- (A e. C -> ([A / x]x e. B <-> A e. B))

Distinct variable group:   x,B

Proof of Theorem sbcel1gv

Step	Hyp	Ref	Expression
1		ax-17 969	. . . 4 |- (y e. B -> A.x y e. B)
2		eleq1 1531	. . . 4 |- (x = y -> (x e. B <-> y e. B))
3	1, 2	sbie 1194	. . 3 |- ([y / x]x e. B <-> y e. B)
4	3	sbcbii 1974	. 2 |- (A e. C -> ([A / y][y / x]x e. B <-> [A / y]y e. B))
5		sbccog 1948	. 2 |- (A e. C -> ([A / y][y / x]x e. B <-> [A / x]x e. B))
6		elisset 1813	. . 3 |- (A e. C -> A e. V)
7		elex 1815	. . . 4 |- (A e. V -> E.y y = A)
8		ax-17 969	. . . . . . . 8 |- (z e. A -> A.y z e. A)
9	8	hbsbc1 1945	. . . . . . 7 |- ((A e. V -> [A / y]y e. B) -> A.y(A e. V -> [A / y]y e. B))
10		ax-17 969	. . . . . . 7 |- ((A e. V -> A e. B) -> A.y(A e. V -> A e. B))
11	9, 10	hbbi 1008	. . . . . 6 |- (((A e. V -> [A / y]y e. B) <-> (A e. V -> A e. B)) -> A.y((A e. V -> [A / y]y e. B) <-> (A e. V -> A e. B)))
12		sbceq1a 1940	. . . . . . . 8 |- (y = A -> (y e. B <-> [A / y]y e. B))
13		eleq1 1531	. . . . . . . 8 |- (y = A -> (y e. B <-> A e. B))
14	12, 13	bitr3d 529	. . . . . . 7 |- (y = A -> ([A / y]y e. B <-> A e. B))
15	14	imbi2d 611	. . . . . 6 |- (y = A -> ((A e. V -> [A / y]y e. B) <-> (A e. V -> A e. B)))
16	11, 15	19.23ai 1062	. . . . 5 |- (E.y y = A -> ((A e. V -> [A / y]y e. B) <-> (A e. V -> A e. B)))
17	16	pm5.74rd 587	. . . 4 |- (E.y y = A -> (A e. V -> ([A / y]y e. B <-> A e. B)))
18	7, 17	mpcom 49	. . 3 |- (A e. V -> ([A / y]y e. B <-> A e. B))
19	6, 18	syl 10	. 2 |- (A e. C -> ([A / y]y e. B <-> A e. B))
20	4, 5, 19	3bitr3d 547	1 |- (A e. C -> ([A / x]x e. B <-> 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:  sbcth2 1978  ra4sbc 1993  tfinds2 3160  dfoprab5 4105  foprab2 4109

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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