Theorem intab 3247

Description: The intersection of a special case of a class abstraction. y may be free in ph and A, which can be thought of a ph(y) and A(y). Typically, abrexex2 4847 or abexssex 4848 can be used to satisfy the second hypothesis.

Hypotheses

Ref	Expression
intab.1	|- A e. _V
intab.2	|- {x | E.y(ph /\ x = A)} e. _V

Assertion

Ref	Expression
intab	|- |^|{x | A.y(ph -> A e. x)} = {x | E.y(ph /\ x = A)}

Distinct variable groups:   x,A   ph,x   x,y

Proof of Theorem intab

Step	Hyp	Ref	Expression
1		eqeq1 1890	. . . . . . . . . . 11 |- (z = x -> (z = A <-> x = A))
2	1	anbi2d 678	. . . . . . . . . 10 |- (z = x -> ((ph /\ z = A) <-> (ph /\ x = A)))
3	2	exbidv 1657	. . . . . . . . 9 |- (z = x -> (E.y(ph /\ z = A) <-> E.y(ph /\ x = A)))
4	3	cbvabv 2420	. . . . . . . 8 |- {z | E.y(ph /\ z = A)} = {x | E.y(ph /\ x = A)}
5		intab.2	. . . . . . . 8 |- {x | E.y(ph /\ x = A)} e. _V
6	4, 5	eqeltri 1967	. . . . . . 7 |- {z | E.y(ph /\ z = A)} e. _V
7		hbe1 1363	. . . . . . . . . 10 |- (E.y(ph /\ z = A) -> A.yE.y(ph /\ z = A))
8	7	hbab 1875	. . . . . . . . 9 |- (x e. {z | E.y(ph /\ z = A)} -> A.y x e. {z | E.y(ph /\ z = A)})
9	8	hbeleq 1997	. . . . . . . 8 |- (x = {z | E.y(ph /\ z = A)} -> A.y x = {z | E.y(ph /\ z = A)})
10		eleq2 1958	. . . . . . . . 9 |- (x = {z | E.y(ph /\ z = A)} -> (A e. x <-> A e. {z | E.y(ph /\ z = A)}))
11	10	imbi2d 674	. . . . . . . 8 |- (x = {z | E.y(ph /\ z = A)} -> ((ph -> A e. x) <-> (ph -> A e. {z | E.y(ph /\ z = A)})))
12	9, 11	albid 1459	. . . . . . 7 |- (x = {z | E.y(ph /\ z = A)} -> (A.y(ph -> A e. x) <-> A.y(ph -> A e. {z | E.y(ph /\ z = A)})))
13	6, 12	sbcie 2485	. . . . . 6 |- ([{z | E.y(ph /\ z = A)} / x]A.y(ph -> A e. x) <-> A.y(ph -> A e. {z | E.y(ph /\ z = A)}))
14		intab.1	. . . . . . . . . . . 12 |- A e. _V
15		ax-17 1317	. . . . . . . . . . . . 13 |- (ph -> A.zph)
16	15	sbcgf 2520	. . . . . . . . . . . 12 |- (A e. _V -> ([A / z]ph <-> ph))
17	14, 16	ax-mp 7	. . . . . . . . . . 11 |- ([A / z]ph <-> ph)
18	17	biimpri 169	. . . . . . . . . 10 |- (ph -> [A / z]ph)
19		csbvarg 2564	. . . . . . . . . . . 12 |- (A e. _V -> [_A / z]_z = A)
20	14, 19	ax-mp 7	. . . . . . . . . . 11 |- [_A / z]_z = A
21		sbceq1dig 2557	. . . . . . . . . . . 12 |- (A e. _V -> ([A / z]z = A <-> [_A / z]_z = A))
22	14, 21	ax-mp 7	. . . . . . . . . . 11 |- ([A / z]z = A <-> [_A / z]_z = A)
23	20, 22	mpbir 207	. . . . . . . . . 10 |- [A / z]z = A
24	18, 23	jctir 317	. . . . . . . . 9 |- (ph -> ([A / z]ph /\ [A / z]z = A))
25		sbcang 2497	. . . . . . . . . 10 |- (A e. _V -> ([A / z](ph /\ z = A) <-> ([A / z]ph /\ [A / z]z = A)))
26	14, 25	ax-mp 7	. . . . . . . . 9 |- ([A / z](ph /\ z = A) <-> ([A / z]ph /\ [A / z]z = A))
27	24, 26	sylibr 217	. . . . . . . 8 |- (ph -> [A / z](ph /\ z = A))
28		19.8a 1376	. . . . . . . . . . 11 |- ((ph /\ z = A) -> E.y(ph /\ z = A))
29	28	ax-gen 1305	. . . . . . . . . 10 |- A.z((ph /\ z = A) -> E.y(ph /\ z = A))
30		a4sbc 2457	. . . . . . . . . 10 |- (A e. _V -> (A.z((ph /\ z = A) -> E.y(ph /\ z = A)) -> [A / z]((ph /\ z = A) -> E.y(ph /\ z = A))))
31	14, 29, 30	mp2 54	. . . . . . . . 9 |- [A / z]((ph /\ z = A) -> E.y(ph /\ z = A))
32		sbcimg 2496	. . . . . . . . . 10 |- (A e. _V -> ([A / z]((ph /\ z = A) -> E.y(ph /\ z = A)) <-> ([A / z](ph /\ z = A) -> [A / z]E.y(ph /\ z = A))))
33	14, 32	ax-mp 7	. . . . . . . . 9 |- ([A / z]((ph /\ z = A) -> E.y(ph /\ z = A)) <-> ([A / z](ph /\ z = A) -> [A / z]E.y(ph /\ z = A)))
34	31, 33	mpbi 206	. . . . . . . 8 |- ([A / z](ph /\ z = A) -> [A / z]E.y(ph /\ z = A))
35	27, 34	syl 12	. . . . . . 7 |- (ph -> [A / z]E.y(ph /\ z = A))
36	14	elabs 2489	. . . . . . 7 |- (A e. {z | E.y(ph /\ z = A)} <-> [A / z]E.y(ph /\ z = A))
37	35, 36	sylibr 217	. . . . . 6 |- (ph -> A e. {z | E.y(ph /\ z = A)})
38	13, 37	mpgbir 1334	. . . . 5 |- [{z | E.y(ph /\ z = A)} / x]A.y(ph -> A e. x)
39	6	elabs 2489	. . . . 5 |- ({z | E.y(ph /\ z = A)} e. {x | A.y(ph -> A e. x)} <-> [{z | E.y(ph /\ z = A)} / x]A.y(ph -> A e. x))
40	38, 39	mpbir 207	. . . 4 |- {z | E.y(ph /\ z = A)} e. {x | A.y(ph -> A e. x)}
41		intss1 3231	. . . 4 |- ({z | E.y(ph /\ z = A)} e. {x | A.y(ph -> A e. x)} -> |^|{x | A.y(ph -> A e. x)} C_ {z | E.y(ph /\ z = A)})
42	40, 41	ax-mp 7	. . 3 |- |^|{x | A.y(ph -> A e. x)} C_ {z | E.y(ph /\ z = A)}
43		hba1 1350	. . . . . . 7 |- (A.y(ph -> A e. x) -> A.yA.y(ph -> A e. x))
44	43	hbab 1875	. . . . . 6 |- (z e. {x | A.y(ph -> A e. x)} -> A.y z e. {x | A.y(ph -> A e. x)})
45	44	hbint 3225	. . . . 5 |- (z e. |^|{x | A.y(ph -> A e. x)} -> A.y z e. |^|{x | A.y(ph -> A e. x)})
46		ax-4 1319	. . . . . . . . . 10 |- (A.y(ph -> A e. x) -> (ph -> A e. x))
47	46	com12 14	. . . . . . . . 9 |- (ph -> (A.y(ph -> A e. x) -> A e. x))
48	47	adantr 425	. . . . . . . 8 |- ((ph /\ z = A) -> (A.y(ph -> A e. x) -> A e. x))
49		eleq1 1957	. . . . . . . . 9 |- (z = A -> (z e. x <-> A e. x))
50	49	adantl 424	. . . . . . . 8 |- ((ph /\ z = A) -> (z e. x <-> A e. x))
51	48, 50	sylibrd 221	. . . . . . 7 |- ((ph /\ z = A) -> (A.y(ph -> A e. x) -> z e. x))
52	51	19.21aiv 1664	. . . . . 6 |- ((ph /\ z = A) -> A.x(A.y(ph -> A e. x) -> z e. x))
53		visset 2295	. . . . . . 7 |- z e. _V
54	53	elintab 3227	. . . . . 6 |- (z e. |^|{x | A.y(ph -> A e. x)} <-> A.x(A.y(ph -> A e. x) -> z e. x))
55	52, 54	sylibr 217	. . . . 5 |- ((ph /\ z = A) -> z e. |^|{x | A.y(ph -> A e. x)})
56	45, 55	19.23ai 1412	. . . 4 |- (E.y(ph /\ z = A) -> z e. |^|{x | A.y(ph -> A e. x)})
57	56	abssi 2682	. . 3 |- {z | E.y(ph /\ z = A)} C_ |^|{x | A.y(ph -> A e. x)}
58	42, 57	eqssi 2632	. 2 |- |^|{x | A.y(ph -> A e. x)} = {z | E.y(ph /\ z = A)}
59	58, 4	eqtri 1908	1 |- |^|{x | A.y(ph -> A e. x)} = {x | E.y(ph /\ x = A)}

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  {cab 1871  _Vcvv 2292  [_csb 2540   C_ wss 2593  |^|cint 3214

This theorem is referenced by:  abfii2 5652

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  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-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-in 2603  df-ss 2605  df-int 3215

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