Theorem intab 3919

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 ). (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

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

Allowed substitution hints:   ph( y)   A( y)

Proof of Theorem intab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2213	. . . . . . . . . 10 |- ( z = x -> ( z = A <-> x = A ) )
2	1	anbi2d 464	. . . . . . . . 9 |- ( z = x -> ( ( ph /\ z = A ) <-> ( ph /\ x = A ) ) )
3	2	exbidv 1849	. . . . . . . 8 |- ( z = x -> ( E. y ( ph /\ z = A ) <-> E. y ( ph /\ x = A ) ) )
4	3	cbvabv 2331	. . . . . . 7 |- { z | E. y ( ph /\ z = A ) } = { x | E. y ( ph /\ x = A ) }
5		intab.2	. . . . . . 7 |- { x | E. y ( ph /\ x = A ) } e. _V
6	4, 5	eqeltri 2279	. . . . . 6 |- { z | E. y ( ph /\ z = A ) } e. _V
7		nfe1 1520	. . . . . . . . 9 |- F/ y E. y ( ph /\ z = A )
8	7	nfab 2354	. . . . . . . 8 |- F/_ y { z | E. y ( ph /\ z = A ) }
9	8	nfeq2 2361	. . . . . . 7 |- F/ y x = { z | E. y ( ph /\ z = A ) }
10		eleq2 2270	. . . . . . . 8 |- ( x = { z | E. y ( ph /\ z = A ) } -> ( A e. x <-> A e. { z | E. y ( ph /\ z = A ) } ) )
11	10	imbi2d 230	. . . . . . 7 |- ( x = { z | E. y ( ph /\ z = A ) } -> ( ( ph -> A e. x ) <-> ( ph -> A e. { z | E. y ( ph /\ z = A ) } ) ) )
12	9, 11	albid 1639	. . . . . 6 |- ( 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	elab 2921	. . . . 5 |- ( { z | E. y ( ph /\ z = A ) } e. { x | A. y ( ph -> A e. x ) } <-> A. y ( ph -> A e. { z | E. y ( ph /\ z = A ) } ) )
14		19.8a 1614	. . . . . . . . 9 |- ( ( ph /\ z = A ) -> E. y ( ph /\ z = A ) )
15	14	ex 115	. . . . . . . 8 |- ( ph -> ( z = A -> E. y ( ph /\ z = A ) ) )
16	15	alrimiv 1898	. . . . . . 7 |- ( ph -> A. z ( z = A -> E. y ( ph /\ z = A ) ) )
17		intab.1	. . . . . . . 8 |- A e. _V
18	17	sbc6 3028	. . . . . . 7 |- ( [. A / z ]. E. y ( ph /\ z = A ) <-> A. z ( z = A -> E. y ( ph /\ z = A ) ) )
19	16, 18	sylibr 134	. . . . . 6 |- ( ph -> [. A / z ]. E. y ( ph /\ z = A ) )
20		df-sbc 3003	. . . . . 6 |- ( [. A / z ]. E. y ( ph /\ z = A ) <-> A e. { z | E. y ( ph /\ z = A ) } )
21	19, 20	sylib 122	. . . . 5 |- ( ph -> A e. { z | E. y ( ph /\ z = A ) } )
22	13, 21	mpgbir 1477	. . . 4 |- { z | E. y ( ph /\ z = A ) } e. { x | A. y ( ph -> A e. x ) }
23		intss1 3905	. . . 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 ) } )
24	22, 23	ax-mp 5	. . 3 |- |^| { x | A. y ( ph -> A e. x ) } C_ { z | E. y ( ph /\ z = A ) }
25		19.29r 1645	. . . . . . . 8 |- ( ( E. y ( ph /\ z = A ) /\ A. y ( ph -> A e. x ) ) -> E. y ( ( ph /\ z = A ) /\ ( ph -> A e. x ) ) )
26		simplr 528	. . . . . . . . . 10 |- ( ( ( ph /\ z = A ) /\ ( ph -> A e. x ) ) -> z = A )
27		pm3.35 347	. . . . . . . . . . 11 |- ( ( ph /\ ( ph -> A e. x ) ) -> A e. x )
28	27	adantlr 477	. . . . . . . . . 10 |- ( ( ( ph /\ z = A ) /\ ( ph -> A e. x ) ) -> A e. x )
29	26, 28	eqeltrd 2283	. . . . . . . . 9 |- ( ( ( ph /\ z = A ) /\ ( ph -> A e. x ) ) -> z e. x )
30	29	exlimiv 1622	. . . . . . . 8 |- ( E. y ( ( ph /\ z = A ) /\ ( ph -> A e. x ) ) -> z e. x )
31	25, 30	syl 14	. . . . . . 7 |- ( ( E. y ( ph /\ z = A ) /\ A. y ( ph -> A e. x ) ) -> z e. x )
32	31	ex 115	. . . . . 6 |- ( E. y ( ph /\ z = A ) -> ( A. y ( ph -> A e. x ) -> z e. x ) )
33	32	alrimiv 1898	. . . . 5 |- ( E. y ( ph /\ z = A ) -> A. x ( A. y ( ph -> A e. x ) -> z e. x ) )
34		vex 2776	. . . . . 6 |- z e. _V
35	34	elintab 3901	. . . . 5 |- ( z e. |^| { x | A. y ( ph -> A e. x ) } <-> A. x ( A. y ( ph -> A e. x ) -> z e. x ) )
36	33, 35	sylibr 134	. . . 4 |- ( E. y ( ph /\ z = A ) -> z e. |^| { x | A. y ( ph -> A e. x ) } )
37	36	abssi 3272	. . 3 |- { z | E. y ( ph /\ z = A ) } C_ |^| { x | A. y ( ph -> A e. x ) }
38	24, 37	eqssi 3213	. 2 |- |^| { x | A. y ( ph -> A e. x ) } = { z | E. y ( ph /\ z = A ) }
39	38, 4	eqtri 2227	1 |- |^| { x | A. y ( ph -> A e. x ) } = { x | E. y ( ph /\ x = A ) }

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   = wceq 1373   E.wex 1516   e. wcel 2177   {cab 2192   _Vcvv 2773   [.wsbc 3002   C_ wss 3170   |^|cint 3890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3003  df-in 3176  df-ss 3183  df-int 3891

This theorem is referenced by: (None)