Theorem inrab2 3237

Description: Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)

Assertion

Ref	Expression
inrab2	|- ( { x e. A | ph } i^i B ) = { x e. ( A i^i B ) | ph }

Distinct variable group:   x, B

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

Proof of Theorem inrab2

Step	Hyp	Ref	Expression
1		df-rab 2357	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		abid2 2199	. . . 4 |- { x | x e. B } = B
3	2	eqcomi 2085	. . 3 |- B = { x | x e. B }
4	1, 3	ineq12i 3165	. 2 |- ( { x e. A | ph } i^i B ) = ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } )
5		df-rab 2357	. . 3 |- { x e. ( A i^i B ) | ph } = { x | ( x e. ( A i^i B ) /\ ph ) }
6		inab 3232	. . . 4 |- ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } ) = { x | ( ( x e. A /\ ph ) /\ x e. B ) }
7		elin 3155	. . . . . . 7 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
8	7	anbi1i 445	. . . . . 6 |- ( ( x e. ( A i^i B ) /\ ph ) <-> ( ( x e. A /\ x e. B ) /\ ph ) )
9		an32 526	. . . . . 6 |- ( ( ( x e. A /\ x e. B ) /\ ph ) <-> ( ( x e. A /\ ph ) /\ x e. B ) )
10	8, 9	bitri 182	. . . . 5 |- ( ( x e. ( A i^i B ) /\ ph ) <-> ( ( x e. A /\ ph ) /\ x e. B ) )
11	10	abbii 2194	. . . 4 |- { x | ( x e. ( A i^i B ) /\ ph ) } = { x | ( ( x e. A /\ ph ) /\ x e. B ) }
12	6, 11	eqtr4i 2104	. . 3 |- ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } ) = { x | ( x e. ( A i^i B ) /\ ph ) }
13	5, 12	eqtr4i 2104	. 2 |- { x e. ( A i^i B ) | ph } = ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } )
14	4, 13	eqtr4i 2104	1 |- ( { x e. A | ph } i^i B ) = { x e. ( A i^i B ) | ph }

Syntax hints:   /\ wa 102   = wceq 1284   e. wcel 1433   {cab 2067   {crab 2352   i^i cin 2972

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-in 2979

This theorem is referenced by:  iooval2  8938  fzval2  9032