Theorem notrab 3214

Description: Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
notrab	|- ( A \ { x e. A | ph } ) = { x e. A | -. ph }

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem notrab

Step	Hyp	Ref	Expression
1		difab 3206	. 2 |- ( { x | x e. A } \ { x | ph } ) = { x | ( x e. A /\ -. ph ) }
2		difin 3174	. . 3 |- ( A \ ( A i^i { x | ph } ) ) = ( A \ { x | ph } )
3		dfrab3 3213	. . . 4 |- { x e. A | ph } = ( A i^i { x | ph } )
4	3	difeq2i 3059	. . 3 |- ( A \ { x e. A | ph } ) = ( A \ ( A i^i { x | ph } ) )
5		abid2 2158	. . . 4 |- { x | x e. A } = A
6	5	difeq1i 3058	. . 3 |- ( { x | x e. A } \ { x | ph } ) = ( A \ { x | ph } )
7	2, 4, 6	3eqtr4i 2070	. 2 |- ( A \ { x e. A | ph } ) = ( { x | x e. A } \ { x | ph } )
8		df-rab 2315	. 2 |- { x e. A | -. ph } = { x | ( x e. A /\ -. ph ) }
9	1, 7, 8	3eqtr4i 2070	1 |- ( A \ { x e. A | ph } ) = { x e. A | -. ph }

Syntax hints:   -. wn 3   /\ wa 97   = wceq 1243   e. wcel 1393   {cab 2026   {crab 2310   \ cdif 2914   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-in1 544  ax-in2 545  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-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-dif 2920  df-in 2924

This theorem is referenced by:  diffitest  6344