Theorem intab 2528

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 3810 or abexssex 3811 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 1457	. . . . . . . . . . 11 |- (z = x -> (z = A <-> x = A))
2	1	anbi2d 614	. . . . . . . . . 10 |- (z = x -> ((ph /\ z = A) <-> (ph /\ x = A)))
3	2	exbidv 1261	. . . . . . . . 9 |- (z = x -> (E.y(ph /\ z = A) <-> E.y(ph /\ x = A)))
4	3	cbvabv 1881	. . . . . . . 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 1520	. . . . . . 7 |- {z | E.y(ph /\ z = A)} e. V
7		hbe1 990	. . . . . . . . . 10 |- (E.y(ph /\ z = A) -> A.yE.y(ph /\ z = A))
8	7	hbab 1444	. . . . . . . . 9 |- (x e. {z | E.y(ph /\ z = A)} -> A.y x e. {z | E.y(ph /\ z = A)})
9	8	hbeleq 1543	. . . . . . . 8 |- (x = {z | E.y(ph /\ z = A)} -> A.y x = {z | E.y(ph /\ z = A)})
10		eleq2 1511	. . . . . . . . 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 1080	. . . . . . 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 1933	. . . . . 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 1190	. . . . . . . . . . . . 13 |- (ph -> A.zph)
16	15	sbcgf 1957	. . . . . . . . . . . 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 1992	. . . . . . . . . . . 12 |- (A e. V -> [_A / z]_z = A)
20	14, 19	ax-mp 7	. . . . . . . . . . 11 |- [_A / z]_z = A
21		sbceq1dig 1985	. . . . . . . . . . . 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 1942	. . . . . . . . . 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 1005	. . . . . . . . . . 11 |- ((ph /\ z = A) -> E.y(ph /\ z = A))
29	28	ax-gen 955	. . . . . . . . . 10 |- A.z((ph /\ z = A) -> E.y(ph /\ z = A))
30		a4sbc 1916	. . . . . . . . . 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 1941	. . . . . . . . . 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 1937	. . . . . . 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 964	. . . . 5 |- [{z | E.y(ph /\ z = A)} / x]A.y(ph -> A e. x)
39	6	elabs 1937	. . . . 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 2516	. . . 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 979	. . . . . . 7 |- (A.y(ph -> A e. x) -> A.yA.y(ph -> A e. x))
44	43	hbab 1444	. . . . . 6 |- (z e. {x | A.y(ph -> A e. x)} -> A.y z e. {x | A.y(ph -> A e. x)})
45	44	hbint 2511	. . . . 5 |- (z e. |^|{x | A.y(ph -> A e. x)} -> A.y z e. |^|{x | A.y(ph -> A e. x)})
46		ax-4 951	. . . . . . . . . 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 1510	. . . . . . . . 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 1268	. . . . . 6 |- ((ph /\ z = A) -> A.x(A.y(ph -> A e. x) -> z e. x))
53		visset 1788	. . . . . . 7 |- z e. V
54	53	elintab 2512	. . . . . 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 1040	. . . 4 |- (E.y(ph /\ z = A) -> z e. |^|{x | A.y(ph -> A e. x)})
57	56	abssi 2093	. . 3 |- {z | E.y(ph /\ z = A)} (_ |^|{x | A.y(ph -> A e. x)}
58	42, 57	eqssi 2049	. 2 |- |^|{x | A.y(ph -> A e. x)} = {z | E.y(ph /\ z = A)}
59	58, 4	eqtr 1471	1 |- |^|{x | A.y(ph -> A e. x)} = {x | E.y(ph /\ x = A)}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105  [wsbc 1153  {cab 1440  Vcvv 1786  [_csb 1972   (_ wss 2018  |^|cint 2501

This theorem is referenced by:  abfii2 4488

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-rab 1628  df-v 1787  df-sbc 1913  df-csb 1973  df-in 2022  df-ss 2024  df-int 2502

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