Theorem notrab 3532

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 3523	. 2 |- ({x | x e. A} \ {x | ph}) = {x | (x e. A /\ -. ph)}
2		difin 3492	. . 3 |- (A \ (A i^i {x | ph})) = (A \ {x | ph})
3		dfrab3 3531	. . . 4 |- {x e. A | ph} = (A i^i {x | ph})
4	3	difeq2i 3382	. . 3 |- (A \ {x e. A | ph}) = (A \ (A i^i {x | ph}))
5		abid2 2470	. . . 4 |- {x | x e. A} = A
6	5	difeq1i 3381	. . 3 |- ({x | x e. A} \ {x | ph}) = (A \ {x | ph})
7	2, 4, 6	3eqtr4i 2383	. 2 |- (A \ {x e. A | ph}) = ({x | x e. A} \ {x | ph})
8		df-rab 2623	. 2 |- {x e. A | -. ph} = {x | (x e. A /\ -. ph)}
9	1, 7, 8	3eqtr4i 2383	1 |- (A \ {x e. A | ph}) = {x e. A | -. ph}

Syntax hints:  -. wn 3   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339  {crab 2618   \ cdif 3206   i^i cin 3208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215

This theorem is referenced by: (None)