Theorem inab 3440

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 1982	. . 3 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
2		df-clab 2191	. . 3 |- ( y e. { x | ( ph /\ ps ) } <-> [ y / x ] ( ph /\ ps ) )
3		df-clab 2191	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4		df-clab 2191	. . . 4 |- ( y e. { x | ps } <-> [ y / x ] ps )
5	3, 4	anbi12i 460	. . 3 |- ( ( y e. { x | ph } /\ y e. { x | ps } ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
6	1, 2, 5	3bitr4ri 213	. 2 |- ( ( y e. { x | ph } /\ y e. { x | ps } ) <-> y e. { x | ( ph /\ ps ) } )
7	6	ineqri 3365	1 |- ( { x | ph } i^i { x | ps } ) = { x | ( ph /\ ps ) }

Syntax hints:   /\ wa 104   = wceq 1372   [wsb 1784   e. wcel 2175   {cab 2190   i^i cin 3164

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171

This theorem is referenced by:  inrab  3444  inrab2  3445  dfrab2  3447  dfrab3  3448  imainlem  5354  imain  5355  ssenen  6947