Theorem difab 3428

Description: Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
difab	|- ( { x | ph } \ { x | ps } ) = { x | ( ph /\ -. ps ) }

Proof of Theorem difab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clab 2180	. . 3 |- ( y e. { x | ( ph /\ -. ps ) } <-> [ y / x ] ( ph /\ -. ps ) )
2		sban 1971	. . 3 |- ( [ y / x ] ( ph /\ -. ps ) <-> ( [ y / x ] ph /\ [ y / x ] -. ps ) )
3		df-clab 2180	. . . . 5 |- ( y e. { x | ph } <-> [ y / x ] ph )
4	3	bicomi 132	. . . 4 |- ( [ y / x ] ph <-> y e. { x | ph } )
5		sbn 1968	. . . . 5 |- ( [ y / x ] -. ps <-> -. [ y / x ] ps )
6		df-clab 2180	. . . . 5 |- ( y e. { x | ps } <-> [ y / x ] ps )
7	5, 6	xchbinxr 684	. . . 4 |- ( [ y / x ] -. ps <-> -. y e. { x | ps } )
8	4, 7	anbi12i 460	. . 3 |- ( ( [ y / x ] ph /\ [ y / x ] -. ps ) <-> ( y e. { x | ph } /\ -. y e. { x | ps } ) )
9	1, 2, 8	3bitrri 207	. 2 |- ( ( y e. { x | ph } /\ -. y e. { x | ps } ) <-> y e. { x | ( ph /\ -. ps ) } )
10	9	difeqri 3279	1 |- ( { x | ph } \ { x | ps } ) = { x | ( ph /\ -. ps ) }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1364   [wsb 1773   e. wcel 2164   {cab 2179   \ cdif 3150

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155

This theorem is referenced by:  notab  3429  difrab  3433  notrab  3436  imadiflem  5333  imadif  5334