Theorem inrab2 3454

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 2495	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		abid2 2328	. . . 4 |- { x | x e. B } = B
3	2	eqcomi 2211	. . 3 |- B = { x | x e. B }
4	1, 3	ineq12i 3380	. 2 |- ( { x e. A | ph } i^i B ) = ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } )
5		df-rab 2495	. . 3 |- { x e. ( A i^i B ) | ph } = { x | ( x e. ( A i^i B ) /\ ph ) }
6		inab 3449	. . . 4 |- ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } ) = { x | ( ( x e. A /\ ph ) /\ x e. B ) }
7		elin 3364	. . . . . . 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 2323	. . . 4 |- { x | ( x e. ( A i^i B ) /\ ph ) } = { x | ( ( x e. A /\ ph ) /\ x e. B ) }
12	6, 11	eqtr4i 2231	. . 3 |- ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } ) = { x | ( x e. ( A i^i B ) /\ ph ) }
13	5, 12	eqtr4i 2231	. 2 |- { x e. ( A i^i B ) | ph } = ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } )
14	4, 13	eqtr4i 2231	1 |- ( { x e. A | ph } i^i B ) = { x e. ( A i^i B ) | ph }

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2178   {cab 2193   {crab 2490   i^i cin 3173

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 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-v 2778  df-in 3180

This theorem is referenced by:  iooval2  10072  fzval2  10168  dfphi2  12657  znnen  12884