Theorem intab 3997

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 2245	. . . . . . . . . 10 |- ( z = x -> ( z = A <-> x = A ) )
2	1	anbi2d 468	. . . . . . . . 9 |- ( z = x -> ( ( ph /\ z = A ) <-> ( ph /\ x = A ) ) )
3	2	exbidv 1878	. . . . . . . 8 |- ( z = x -> ( E. y ( ph /\ z = A ) <-> E. y ( ph /\ x = A ) ) )
4	3	cbvabv 2365	. . . . . . 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 2311	. . . . . 6 |- { z | E. y ( ph /\ z = A ) } e. _V
7		nfe1 1549	. . . . . . . . 9 |- F/ y E. y ( ph /\ z = A )
8	7	nfab 2397	. . . . . . . 8 |- F/_ y { z | E. y ( ph /\ z = A ) }
9	8	nfeq2 2404	. . . . . . 7 |- F/ y x = { z | E. y ( ph /\ z = A ) }
10		eleq2 2302	. . . . . . . 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 1668	. . . . . 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 2970	. . . . 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 1643	. . . . . . . . 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 1927	. . . . . . 7 |- ( ph -> A. z ( z = A -> E. y ( ph /\ z = A ) ) )
17		intab.1	. . . . . . . 8 |- A e. _V
18	17	sbc6 3077	. . . . . . 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 3052	. . . . . 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 1506	. . . 4 |- { z | E. y ( ph /\ z = A ) } e. { x | A. y ( ph -> A e. x ) }
23		intss1 3983	. . . 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 1674	. . . . . . . 8 |- ( ( E. y ( ph /\ z = A ) /\ A. y ( ph -> A e. x ) ) -> E. y ( ( ph /\ z = A ) /\ ( ph -> A e. x ) ) )
26		simplr 533	. . . . . . . . . 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 481	. . . . . . . . . 10 |- ( ( ( ph /\ z = A ) /\ ( ph -> A e. x ) ) -> A e. x )
29	26, 28	eqeltrd 2315	. . . . . . . . 9 |- ( ( ( ph /\ z = A ) /\ ( ph -> A e. x ) ) -> z e. x )
30	29	exlimiv 1651	. . . . . . . 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 1927	. . . . 5 |- ( E. y ( ph /\ z = A ) -> A. x ( A. y ( ph -> A e. x ) -> z e. x ) )
34		vex 2824	. . . . . 6 |- z e. _V
35	34	elintab 3979	. . . . 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 3323	. . 3 |- { z | E. y ( ph /\ z = A ) } C_ |^| { x | A. y ( ph -> A e. x ) }
38	24, 37	eqssi 3264	. 2 |- |^| { x | A. y ( ph -> A e. x ) } = { z | E. y ( ph /\ z = A ) }
39	38, 4	eqtri 2259	1 |- |^| { x | A. y ( ph -> A e. x ) } = { x | E. y ( ph /\ x = A ) }

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1400   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   _Vcvv 2821   [.wsbc 3051   C_ wss 3220   |^|cint 3968

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052  df-in 3226  df-ss 3233  df-int 3969

This theorem is referenced by: (None)