Theorem intab 2564

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 3877 or abexssex 3878 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 1484	. . . . . . . . . . 11 |- (z = x -> (z = A <-> x = A))
2	1	anbi2d 618	. . . . . . . . . 10 |- (z = x -> ((ph /\ z = A) <-> (ph /\ x = A)))
3	2	exbidv 1281	. . . . . . . . 9 |- (z = x -> (E.y(ph /\ z = A) <-> E.y(ph /\ x = A)))
4	3	cbvabv 1912	. . . . . . . 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 1547	. . . . . . 7 |- {z | E.y(ph /\ z = A)} e. V
7		hbe1 1018	. . . . . . . . . 10 |- (E.y(ph /\ z = A) -> A.yE.y(ph /\ z = A))
8	7	hbab 1470	. . . . . . . . 9 |- (x e. {z | E.y(ph /\ z = A)} -> A.y x e. {z | E.y(ph /\ z = A)})
9	8	hbeleq 1570	. . . . . . . 8 |- (x = {z | E.y(ph /\ z = A)} -> A.y x = {z | E.y(ph /\ z = A)})
10		eleq2 1538	. . . . . . . . 9 |- (x = {z | E.y(ph /\ z = A)} -> (A e. x <-> A e. {z | E.y(ph /\ z = A)}))
11	10	imbi2d 614	. . . . . . . 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 1106	. . . . . . 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 1965	. . . . . 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-17 973	. . . . . . . . . . . . 13 |- (ph -> A.zph)
16	15	sbcgf 1989	. . . . . . . . . . . 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 2024	. . . . . . . . . . . 12 |- (A e. V -> [_A / z]_z = A)
20	14, 19	ax-mp 7	. . . . . . . . . . 11 |- [_A / z]_z = A
21		sbceq1dig 2017	. . . . . . . . . . . 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 1974	. . . . . . . . . 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 1031	. . . . . . . . . . 11 |- ((ph /\ z = A) -> E.y(ph /\ z = A))
29	28	ax-gen 965	. . . . . . . . . 10 |- A.z((ph /\ z = A) -> E.y(ph /\ z = A))
30		a4sbc 1948	. . . . . . . . . 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 1973	. . . . . . . . . 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 1969	. . . . . . 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 990	. . . . 5 |- [{z | E.y(ph /\ z = A)} / x]A.y(ph -> A e. x)
39	6	elabs 1969	. . . . 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 2552	. . . 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 1005	. . . . . . 7 |- (A.y(ph -> A e. x) -> A.yA.y(ph -> A e. x))
44	43	hbab 1470	. . . . . 6 |- (z e. {x | A.y(ph -> A e. x)} -> A.y z e. {x | A.y(ph -> A e. x)})
45	44	hbint 2547	. . . . 5 |- (z e. |^|{x | A.y(ph -> A e. x)} -> A.y z e. |^|{x | A.y(ph -> A e. x)})
46		ax-4 975	. . . . . . . . . 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 391	. . . . . . . 8 |- ((ph /\ z = A) -> (A.y(ph -> A e. x) -> A e. x))
49		eleq1 1537	. . . . . . . . 9 |- (z = A -> (z e. x <-> A e. x))
50	49	adantl 390	. . . . . . . 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 1288	. . . . . 6 |- ((ph /\ z = A) -> A.x(A.y(ph -> A e. x) -> z e. x))
53		visset 1816	. . . . . . 7 |- z e. V
54	53	elintab 2548	. . . . . 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 1066	. . . 4 |- (E.y(ph /\ z = A) -> z e. |^|{x | A.y(ph -> A e. x)})
57	56	abssi 2125	. . 3 |- {z | E.y(ph /\ z = A)} (_ |^|{x | A.y(ph -> A e. x)}
58	42, 57	eqssi 2081	. 2 |- |^|{x | A.y(ph -> A e. x)} = {z | E.y(ph /\ z = A)}
59	58, 4	eqtr 1498	1 |- |^|{x | A.y(ph -> A e. x)} = {x | E.y(ph /\ x = A)}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  [wsbc 1172  {cab 1466  Vcvv 1814  [_csb 2004   (_ wss 2050  |^|cint 2537

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-in 2054  df-ss 2056  df-int 2538

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