Theorem sbcralgf 1988

Description: Interchange class substitution and restricted quantifier.

Hypothesis

Ref	Expression
sbcralgf.1	|- (A.y A e. C -> (z e. A -> A.y z e. A))

Assertion

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

Distinct variable groups:   z,A   x,B   z,C   x,y,z

Proof of Theorem sbcralgf

Step	Hyp	Ref	Expression
1		sbc6g 1951	. . . . 5 |- (A e. C -> ([A / w]A.y e. B [w / x]ph <-> A.w(w = A -> A.y e. B [w / x]ph)))
2	1	a4s 982	. . . 4 |- (A.y A e. C -> ([A / w]A.y e. B [w / x]ph <-> A.w(w = A -> A.y e. B [w / x]ph)))
3		hba1 1001	. . . . . . . . 9 |- (A.y A e. C -> A.yA.y A e. C)
4		ax-17 969	. . . . . . . . . 10 |- (z e. w -> A.y z e. w)
5	4	a1i 8	. . . . . . . . 9 |- (A.y A e. C -> (z e. w -> A.y z e. w))
6		sbcralgf.1	. . . . . . . . 9 |- (A.y A e. C -> (z e. A -> A.y z e. A))
7	3, 5, 6	hbeqd 1909	. . . . . . . 8 |- (A.y A e. C -> (w = A -> A.y w = A))
8	7	a5i 987	. . . . . . 7 |- (A.y A e. C -> A.y(w = A -> A.y w = A))
9		r19.21t 1712	. . . . . . 7 |- (A.y(w = A -> A.y w = A) -> (A.y e. B (w = A -> [w / x]ph) <-> (w = A -> A.y e. B [w / x]ph)))
10	8, 9	syl 10	. . . . . 6 |- (A.y A e. C -> (A.y e. B (w = A -> [w / x]ph) <-> (w = A -> A.y e. B [w / x]ph)))
11	10	albidv 1276	. . . . 5 |- (A.y A e. C -> (A.wA.y e. B (w = A -> [w / x]ph) <-> A.w(w = A -> A.y e. B [w / x]ph)))
12		ralcom4 1819	. . . . 5 |- (A.y e. B A.w(w = A -> [w / x]ph) <-> A.wA.y e. B (w = A -> [w / x]ph))
13	11, 12	syl5rbb 532	. . . 4 |- (A.y A e. C -> (A.w(w = A -> A.y e. B [w / x]ph) <-> A.y e. B A.w(w = A -> [w / x]ph)))
14	2, 13	bitrd 527	. . 3 |- (A.y A e. C -> ([A / w]A.y e. B [w / x]ph <-> A.y e. B A.w(w = A -> [w / x]ph)))
15		visset 1809	. . . . . 6 |- w e. V
16		sbc6g 1951	. . . . . . . 8 |- (w e. V -> ([w / x]A.y e. B ph <-> A.x(x = w -> A.y e. B ph)))
17		ralcom4 1819	. . . . . . . . 9 |- (A.y e. B A.x(x = w -> ph) <-> A.xA.y e. B (x = w -> ph))
18		r19.21v 1713	. . . . . . . . . 10 |- (A.y e. B (x = w -> ph) <-> (x = w -> A.y e. B ph))
19	18	albii 997	. . . . . . . . 9 |- (A.xA.y e. B (x = w -> ph) <-> A.x(x = w -> A.y e. B ph))
20	17, 19	bitr2 174	. . . . . . . 8 |- (A.x(x = w -> A.y e. B ph) <-> A.y e. B A.x(x = w -> ph))
21	16, 20	syl6bb 535	. . . . . . 7 |- (w e. V -> ([w / x]A.y e. B ph <-> A.y e. B A.x(x = w -> ph)))
22		sbc6g 1951	. . . . . . . 8 |- (w e. V -> ([w / x]ph <-> A.x(x = w -> ph)))
23	22	ralbidv 1660	. . . . . . 7 |- (w e. V -> (A.y e. B [w / x]ph <-> A.y e. B A.x(x = w -> ph)))
24	21, 23	bitr4d 530	. . . . . 6 |- (w e. V -> ([w / x]A.y e. B ph <-> A.y e. B [w / x]ph))
25	15, 24	ax-mp 7	. . . . 5 |- ([w / x]A.y e. B ph <-> A.y e. B [w / x]ph)
26	25	sbcbii 1974	. . . 4 |- (A e. C -> ([A / w][w / x]A.y e. B ph <-> [A / w]A.y e. B [w / x]ph))
27	26	a4s 982	. . 3 |- (A.y A e. C -> ([A / w][w / x]A.y e. B ph <-> [A / w]A.y e. B [w / x]ph))
28		sbc6g 1951	. . . . 5 |- (A e. C -> ([A / w][w / x]ph <-> A.w(w = A -> [w / x]ph)))
29	28	a4s 982	. . . 4 |- (A.y A e. C -> ([A / w][w / x]ph <-> A.w(w = A -> [w / x]ph)))
30	3, 29	ralbid 1658	. . 3 |- (A.y A e. C -> (A.y e. B [A / w][w / x]ph <-> A.y e. B A.w(w = A -> [w / x]ph)))
31	14, 27, 30	3bitr4d 549	. 2 |- (A.y A e. C -> ([A / w][w / x]A.y e. B ph <-> A.y e. B [A / w][w / x]ph))
32		sbccog 1948	. . 3 |- (A e. C -> ([A / w][w / x]A.y e. B ph <-> [A / x]A.y e. B ph))
33	32	a4s 982	. 2 |- (A.y A e. C -> ([A / w][w / x]A.y e. B ph <-> [A / x]A.y e. B ph))
34		sbccog 1948	. . . 4 |- (A e. C -> ([A / w][w / x]ph <-> [A / x]ph))
35	34	a4s 982	. . 3 |- (A.y A e. C -> ([A / w][w / x]ph <-> [A / x]ph))
36	3, 35	ralbid 1658	. 2 |- (A.y A e. C -> (A.y e. B [A / w][w / x]ph <-> A.y e. B [A / x]ph))
37	31, 33, 36	3bitr3d 547	1 |- (A.y A e. C -> ([A / x]A.y e. B ph <-> A.y e. B [A / x]ph))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952   = wceq 954   e. wcel 956  [wsbc 1168  A.wral 1642  Vcvv 1807

This theorem is referenced by:  sbcrexgf 1989

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-ral 1646  df-v 1808  df-sbc 1938

Copyright terms: Public domain