Theorem inrab2 3210

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 2315	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		abid2 2158	. . . 4 |- { x | x e. B } = B
3	2	eqcomi 2044	. . 3 |- B = { x | x e. B }
4	1, 3	ineq12i 3136	. 2 |- ( { x e. A | ph } i^i B ) = ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } )
5		df-rab 2315	. . 3 |- { x e. ( A i^i B ) | ph } = { x | ( x e. ( A i^i B ) /\ ph ) }
6		inab 3205	. . . 4 |- ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } ) = { x | ( ( x e. A /\ ph ) /\ x e. B ) }
7		elin 3126	. . . . . . 7 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
8	7	anbi1i 431	. . . . . 6 |- ( ( x e. ( A i^i B ) /\ ph ) <-> ( ( x e. A /\ x e. B ) /\ ph ) )
9		an32 496	. . . . . 6 |- ( ( ( x e. A /\ x e. B ) /\ ph ) <-> ( ( x e. A /\ ph ) /\ x e. B ) )
10	8, 9	bitri 173	. . . . 5 |- ( ( x e. ( A i^i B ) /\ ph ) <-> ( ( x e. A /\ ph ) /\ x e. B ) )
11	10	abbii 2153	. . . 4 |- { x | ( x e. ( A i^i B ) /\ ph ) } = { x | ( ( x e. A /\ ph ) /\ x e. B ) }
12	6, 11	eqtr4i 2063	. . 3 |- ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } ) = { x | ( x e. ( A i^i B ) /\ ph ) }
13	5, 12	eqtr4i 2063	. 2 |- { x e. ( A i^i B ) | ph } = ( { x | ( x e. A /\ ph ) } i^i { x | x e. B } )
14	4, 13	eqtr4i 2063	1 |- ( { x e. A | ph } i^i B ) = { x e. ( A i^i B ) | ph }

Syntax hints:   /\ wa 97   = wceq 1243   e. wcel 1393   {cab 2026   {crab 2310   i^i cin 2916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-in 2924

This theorem is referenced by:  iooval2  8784  fzval2  8877