Theorem difrab 3530

Description: Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)

Assertion

Ref	Expression
difrab	|- ({x e. A | ph} \ {x e. A | ps}) = {x e. A | (ph /\ -. ps)}

Proof of Theorem difrab

Step	Hyp	Ref	Expression
1		df-rab 2624	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
2		df-rab 2624	. . 3 |- {x e. A | ps} = {x | (x e. A /\ ps)}
3	1, 2	difeq12i 3384	. 2 |- ({x e. A | ph} \ {x e. A | ps}) = ({x | (x e. A /\ ph)} \ {x | (x e. A /\ ps)})
4		df-rab 2624	. . 3 |- {x e. A | (ph /\ -. ps)} = {x | (x e. A /\ (ph /\ -. ps))}
5		difab 3524	. . . 4 |- ({x | (x e. A /\ ph)} \ {x | (x e. A /\ ps)}) = {x | ((x e. A /\ ph) /\ -. (x e. A /\ ps))}
6		anass 630	. . . . . 6 |- (((x e. A /\ ph) /\ -. ps) <-> (x e. A /\ (ph /\ -. ps)))
7		simpr 447	. . . . . . . . 9 |- ((x e. A /\ ps) -> ps)
8	7	con3i 127	. . . . . . . 8 |- (-. ps -> -. (x e. A /\ ps))
9	8	anim2i 552	. . . . . . 7 |- (((x e. A /\ ph) /\ -. ps) -> ((x e. A /\ ph) /\ -. (x e. A /\ ps)))
10		pm3.2 434	. . . . . . . . . 10 |- (x e. A -> (ps -> (x e. A /\ ps)))
11	10	adantr 451	. . . . . . . . 9 |- ((x e. A /\ ph) -> (ps -> (x e. A /\ ps)))
12	11	con3d 125	. . . . . . . 8 |- ((x e. A /\ ph) -> (-. (x e. A /\ ps) -> -. ps))
13	12	imdistani 671	. . . . . . 7 |- (((x e. A /\ ph) /\ -. (x e. A /\ ps)) -> ((x e. A /\ ph) /\ -. ps))
14	9, 13	impbii 180	. . . . . 6 |- (((x e. A /\ ph) /\ -. ps) <-> ((x e. A /\ ph) /\ -. (x e. A /\ ps)))
15	6, 14	bitr3i 242	. . . . 5 |- ((x e. A /\ (ph /\ -. ps)) <-> ((x e. A /\ ph) /\ -. (x e. A /\ ps)))
16	15	abbii 2466	. . . 4 |- {x | (x e. A /\ (ph /\ -. ps))} = {x | ((x e. A /\ ph) /\ -. (x e. A /\ ps))}
17	5, 16	eqtr4i 2376	. . 3 |- ({x | (x e. A /\ ph)} \ {x | (x e. A /\ ps)}) = {x | (x e. A /\ (ph /\ -. ps))}
18	4, 17	eqtr4i 2376	. 2 |- {x e. A | (ph /\ -. ps)} = ({x | (x e. A /\ ph)} \ {x | (x e. A /\ ps)})
19	3, 18	eqtr4i 2376	1 |- ({x e. A | ph} \ {x e. A | ps}) = {x e. A | (ph /\ -. ps)}

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339  {crab 2619   \ cdif 3207

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-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216

This theorem is referenced by: (None)