Theorem inab 3390

Description: Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
inab	|- ( { x | ph } i^i { x | ps } ) = { x | ( ph /\ ps ) }

Proof of Theorem inab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sban 1943	. . 3 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
2		df-clab 2152	. . 3 |- ( y e. { x | ( ph /\ ps ) } <-> [ y / x ] ( ph /\ ps ) )
3		df-clab 2152	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4		df-clab 2152	. . . 4 |- ( y e. { x | ps } <-> [ y / x ] ps )
5	3, 4	anbi12i 456	. . 3 |- ( ( y e. { x | ph } /\ y e. { x | ps } ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
6	1, 2, 5	3bitr4ri 212	. 2 |- ( ( y e. { x | ph } /\ y e. { x | ps } ) <-> y e. { x | ( ph /\ ps ) } )
7	6	ineqri 3315	1 |- ( { x | ph } i^i { x | ps } ) = { x | ( ph /\ ps ) }

Syntax hints:   /\ wa 103   = wceq 1343   [wsb 1750   e. wcel 2136   {cab 2151   i^i cin 3115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122

This theorem is referenced by:  inrab  3394  inrab2  3395  dfrab2  3397  dfrab3  3398  imainlem  5269  imain  5270  ssenen  6817