Theorem complab 3525

Description: Complement of a class abstraction. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
complab	|- ∼ {x | ph} = {x | -. ph}

Proof of Theorem complab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clab 2340	. . . . 5 |- (y e. {x | ph} <-> [y / x]ph)
2	1	notbii 287	. . . 4 |- (-. y e. {x | ph} <-> -. [y / x]ph)
3		sbn 2062	. . . 4 |- ([y / x] -. ph <-> -. [y / x]ph)
4	2, 3	bitr4i 243	. . 3 |- (-. y e. {x | ph} <-> [y / x] -. ph)
5		vex 2863	. . . 4 |- y e. _V
6	5	elcompl 3226	. . 3 |- (y e. ∼ {x | ph} <-> -. y e. {x | ph})
7		df-clab 2340	. . 3 |- (y e. {x | -. ph} <-> [y / x] -. ph)
8	4, 6, 7	3bitr4i 268	. 2 |- (y e. ∼ {x | ph} <-> y e. {x | -. ph})
9	8	eqriv 2350	1 |- ∼ {x | ph} = {x | -. ph}

Syntax hints:  -. wn 3   = wceq 1642  [wsb 1648   e. wcel 1710  {cab 2339   ∼ ccompl 3206

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 2479  df-v 2862  df-nin 3212  df-compl 3213

This theorem is referenced by:  nulnnn  4557  addccan2nclem2  6265