Theorem intab 2550

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 3856 or abexssex 3857 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 1473	. . . . . . . . . . 11 |- (z = x -> (z = A <-> x = A))
2	1	anbi2d 614	. . . . . . . . . 10 |- (z = x -> ((ph /\ z = A) <-> (ph /\ x = A)))
3	2	exbidv 1274	. . . . . . . . 9 |- (z = x -> (E.y(ph /\ z = A) <-> E.y(ph /\ x = A)))
4	3	cbvabv 1900	. . . . . . . 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	eqeltr 1536	. . . . . . 7 |- {z | E.y(ph /\ z = A)} e. V
7		hbe1 1012	. . . . . . . . . 10 |- (E.y(ph /\ z = A) -> A.yE.y(ph /\ z = A))
8	7	hbab 1460	. . . . . . . . 9 |- (x e. {z | E.y(ph /\ z = A)} -> A.y x e. {z | E.y(ph /\ z = A)})
9	8	hbeleq 1559	. . . . . . . 8 |- (x = {z | E.y(ph /\ z = A)} -> A.y x = {z | E.y(ph /\ z = A)})
10		eleq2 1527	. . . . . . . . 9 |- (x = {z | E.y(ph /\ z = A)} -> (A e. x <-> A e. {z | E.y(ph /\ z = A)}))
11	10	imbi2d 610	. . . . . . . 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 1100	. . . . . . 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 1952	. . . . . 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-17 968	. . . . . . . . . . . . 13 |- (ph -> A.zph)
16	15	sbcgf 1976	. . . . . . . . . . . 12 |- (A e. V -> ([A / z]ph <-> ph))
17	14, 16	ax-mp 7	. . . . . . . . . . 11 |- ([A / z]ph <-> ph)
18	17	biimpr 152	. . . . . . . . . 10 |- (ph -> [A / z]ph)
19		csbvarg 2011	. . . . . . . . . . . 12 |- (A e. V -> [_A / z]_z = A)
20	14, 19	ax-mp 7	. . . . . . . . . . 11 |- [_A / z]_z = A
21		sbceq1dig 2004	. . . . . . . . . . . 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 190	. . . . . . . . . 10 |- [A / z]z = A
24	18, 23	jctir 293	. . . . . . . . 9 |- (ph -> ([A / z]ph /\ [A / z]z = A))
25		sbcang 1961	. . . . . . . . . 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 200	. . . . . . . 8 |- (ph -> [A / z](ph /\ z = A))
28		19.8a 1025	. . . . . . . . . . 11 |- ((ph /\ z = A) -> E.y(ph /\ z = A))
29	28	ax-gen 960	. . . . . . . . . 10 |- A.z((ph /\ z = A) -> E.y(ph /\ z = A))
30		a4sbc 1935	. . . . . . . . . 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 43	. . . . . . . . 9 |- [A / z]((ph /\ z = A) -> E.y(ph /\ z = A))
32		sbcimg 1960	. . . . . . . . . 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 189	. . . . . . . 8 |- ([A / z](ph /\ z = A) -> [A / z]E.y(ph /\ z = A))
35	27, 34	syl 10	. . . . . . 7 |- (ph -> [A / z]E.y(ph /\ z = A))
36	14	elabs 1956	. . . . . . 7 |- (A e. {z | E.y(ph /\ z = A)} <-> [A / z]E.y(ph /\ z = A))
37	35, 36	sylibr 200	. . . . . 6 |- (ph -> A e. {z | E.y(ph /\ z = A)})
38	13, 37	mpgbir 985	. . . . 5 |- [{z | E.y(ph /\ z = A)} / x]A.y(ph -> A e. x)
39	6	elabs 1956	. . . . 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 190	. . . 4 |- {z | E.y(ph /\ z = A)} e. {x | A.y(ph -> A e. x)}
41		intss1 2538	. . . 4 |- ({z | E.y(ph /\ z = A)} e. {x | A.y(ph -> A e. x)} -> |^|{x | A.y(ph -> A e. x)} (_ {z | E.y(ph /\ z = A)})
42	40, 41	ax-mp 7	. . 3 |- |^|{x | A.y(ph -> A e. x)} (_ {z | E.y(ph /\ z = A)}
43		hba1 1000	. . . . . . 7 |- (A.y(ph -> A e. x) -> A.yA.y(ph -> A e. x))
44	43	hbab 1460	. . . . . 6 |- (z e. {x | A.y(ph -> A e. x)} -> A.y z e. {x | A.y(ph -> A e. x)})
45	44	hbint 2533	. . . . 5 |- (z e. |^|{x | A.y(ph -> A e. x)} -> A.y z e. |^|{x | A.y(ph -> A e. x)})
46		ax-4 970	. . . . . . . . . 10 |- (A.y(ph -> A e. x) -> (ph -> A e. x))
47	46	com12 11	. . . . . . . . 9 |- (ph -> (A.y(ph -> A e. x) -> A e. x))
48	47	adantr 389	. . . . . . . 8 |- ((ph /\ z = A) -> (A.y(ph -> A e. x) -> A e. x))
49		eleq1 1526	. . . . . . . . 9 |- (z = A -> (z e. x <-> A e. x))
50	49	adantl 388	. . . . . . . 8 |- ((ph /\ z = A) -> (z e. x <-> A e. x))
51	48, 50	sylibrd 204	. . . . . . 7 |- ((ph /\ z = A) -> (A.y(ph -> A e. x) -> z e. x))
52	51	19.21aiv 1281	. . . . . 6 |- ((ph /\ z = A) -> A.x(A.y(ph -> A e. x) -> z e. x))
53		visset 1804	. . . . . . 7 |- z e. V
54	53	elintab 2534	. . . . . 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 200	. . . . 5 |- ((ph /\ z = A) -> z e. |^|{x | A.y(ph -> A e. x)})
56	45, 55	19.23ai 1060	. . . 4 |- (E.y(ph /\ z = A) -> z e. |^|{x | A.y(ph -> A e. x)})
57	56	abssi 2112	. . 3 |- {z | E.y(ph /\ z = A)} (_ |^|{x | A.y(ph -> A e. x)}
58	42, 57	eqssi 2068	. 2 |- |^|{x | A.y(ph -> A e. x)} = {z | E.y(ph /\ z = A)}
59	58, 4	eqtr 1487	1 |- |^|{x | A.y(ph -> A e. x)} = {x | E.y(ph /\ x = A)}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  [wsbc 1166  {cab 1456  Vcvv 1802  [_csb 1991   (_ wss 2037  |^|cint 2523

This theorem is referenced by:  abfii2 4536

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-in 2041  df-ss 2043  df-int 2524

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