Theorem sbcralgf 2529

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 2472	. . . . 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 1330	. . . 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 1350	. . . . . . . . 9 |- (A.y A e. C -> A.yA.y A e. C)
4		ax-17 1317	. . . . . . . . . 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 2424	. . . . . . . 8 |- (A.y A e. C -> (w = A -> A.y w = A))
8	7	a5i 1335	. . . . . . 7 |- (A.y A e. C -> A.y(w = A -> A.y w = A))
9		r19.21t 2177	. . . . . . 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 12	. . . . . 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 1656	. . . . 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 2310	. . . . 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 592	. . . 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 587	. . 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 2295	. . . . . 6 |- w e. _V
16		sbc6g 2472	. . . . . . . 8 |- (w e. _V -> ([w / x]A.y e. B ph <-> A.x(x = w -> A.y e. B ph)))
17		ralcom4 2310	. . . . . . . . 9 |- (A.y e. B A.x(x = w -> ph) <-> A.xA.y e. B (x = w -> ph))
18		r19.21v 2178	. . . . . . . . . 10 |- (A.y e. B (x = w -> ph) <-> (x = w -> A.y e. B ph))
19	18	albii 1346	. . . . . . . . 9 |- (A.xA.y e. B (x = w -> ph) <-> A.x(x = w -> A.y e. B ph))
20	17, 19	bitr2i 191	. . . . . . . 8 |- (A.x(x = w -> A.y e. B ph) <-> A.y e. B A.x(x = w -> ph))
21	16, 20	syl6bb 595	. . . . . . 7 |- (w e. _V -> ([w / x]A.y e. B ph <-> A.y e. B A.x(x = w -> ph)))
22		sbc6g 2472	. . . . . . . 8 |- (w e. _V -> ([w / x]ph <-> A.x(x = w -> ph)))
23	22	ralbidv 2123	. . . . . . 7 |- (w e. _V -> (A.y e. B [w / x]ph <-> A.y e. B A.x(x = w -> ph)))
24	21, 23	bitr4d 590	. . . . . 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 2506	. . . 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 1330	. . 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 2472	. . . . 5 |- (A e. C -> ([A / w][w / x]ph <-> A.w(w = A -> [w / x]ph)))
29	28	a4s 1330	. . . 4 |- (A.y A e. C -> ([A / w][w / x]ph <-> A.w(w = A -> [w / x]ph)))
30	3, 29	ralbid 2121	. . 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 609	. 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 2467	. . 3 |- (A e. C -> ([A / w][w / x]A.y e. B ph <-> [A / x]A.y e. B ph))
33	32	a4s 1330	. 2 |- (A.y A e. C -> ([A / w][w / x]A.y e. B ph <-> [A / x]A.y e. B ph))
34		sbccog 2467	. . . 4 |- (A e. C -> ([A / w][w / x]ph <-> [A / x]ph))
35	34	a4s 1330	. . 3 |- (A.y A e. C -> ([A / w][w / x]ph <-> [A / x]ph))
36	3, 35	ralbid 2121	. 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 607	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 163  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  _Vcvv 2292

This theorem is referenced by:  sbcrexgf 2530

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-sbc 2454

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