Theorem inrab2 3477

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 2517	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		abid2 2350	. . . 4 |- { x | x e. B } = B
3	2	eqcomi 2233	. . 3 |- B = { x | x e. B }
4	1, 3	ineq12i 3403	. 2 |- ( { x e. A | ph } i^i B ) = ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } )
5		df-rab 2517	. . 3 |- { x e. ( A i^i B ) | ph } = { x | ( x e. ( A i^i B ) /\ ph ) }
6		inab 3472	. . . 4 |- ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } ) = { x | ( ( x e. A /\ ph ) /\ x e. B ) }
7		elin 3387	. . . . . . 7 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
8	7	anbi1i 458	. . . . . 6 |- ( ( x e. ( A i^i B ) /\ ph ) <-> ( ( x e. A /\ x e. B ) /\ ph ) )
9		an32 562	. . . . . 6 |- ( ( ( x e. A /\ x e. B ) /\ ph ) <-> ( ( x e. A /\ ph ) /\ x e. B ) )
10	8, 9	bitri 184	. . . . 5 |- ( ( x e. ( A i^i B ) /\ ph ) <-> ( ( x e. A /\ ph ) /\ x e. B ) )
11	10	abbii 2345	. . . 4 |- { x | ( x e. ( A i^i B ) /\ ph ) } = { x | ( ( x e. A /\ ph ) /\ x e. B ) }
12	6, 11	eqtr4i 2253	. . 3 |- ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } ) = { x | ( x e. ( A i^i B ) /\ ph ) }
13	5, 12	eqtr4i 2253	. 2 |- { x e. ( A i^i B ) | ph } = ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } )
14	4, 13	eqtr4i 2253	1 |- ( { x e. A | ph } i^i B ) = { x e. ( A i^i B ) | ph }

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   {cab 2215   {crab 2512   i^i cin 3196

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-in 3203

This theorem is referenced by:  iooval2  10111  fzval2  10207  dfphi2  12742  znnen  12969