Theorem inab 3445

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 1984	. . 3 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
2		df-clab 2193	. . 3 |- ( y e. { x | ( ph /\ ps ) } <-> [ y / x ] ( ph /\ ps ) )
3		df-clab 2193	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
4		df-clab 2193	. . . 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 3370	1 |- ( { x | ph } i^i { x | ps } ) = { x | ( ph /\ ps ) }

Syntax hints:   /\ wa 104   = wceq 1373   [wsb 1786   e. wcel 2177   {cab 2192   i^i cin 3169

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 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176

This theorem is referenced by:  inrab  3449  inrab2  3450  dfrab2  3452  dfrab3  3453  imainlem  5364  imain  5365  ssenen  6963