Theorem difrab 1695

Description: Difference of two restricted class abstractions.

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		difab 1693	. . 3 |- ({x | (x e. A /\ ph)} \ {x | (x e. A /\ ps)}) = {x | ((x e. A /\ ph) /\ -. (x e. A /\ ps))}
2		anass 336	. . . . 5 |- (((x e. A /\ ph) /\ -. ps) <-> (x e. A /\ (ph /\ -. ps)))
3		pm3.27 260	. . . . . . . 8 |- ((x e. A /\ ps) -> ps)
4	3	con3i 90	. . . . . . 7 |- (-. ps -> -. (x e. A /\ ps))
5	4	anim2i 270	. . . . . 6 |- (((x e. A /\ ph) /\ -. ps) -> ((x e. A /\ ph) /\ -. (x e. A /\ ps)))
6		pm3.2 232	. . . . . . . . 9 |- (x e. A -> (ps -> (x e. A /\ ps)))
7	6	adantr 306	. . . . . . . 8 |- ((x e. A /\ ph) -> (ps -> (x e. A /\ ps)))
8	7	con3d 87	. . . . . . 7 |- ((x e. A /\ ph) -> (-. (x e. A /\ ps) -> -. ps))
9	8	imdistani 340	. . . . . 6 |- (((x e. A /\ ph) /\ -. (x e. A /\ ps)) -> ((x e. A /\ ph) /\ -. ps))
10	5, 9	impbi 139	. . . . 5 |- (((x e. A /\ ph) /\ -. ps) <-> ((x e. A /\ ph) /\ -. (x e. A /\ ps)))
11	2, 10	bitr3 153	. . . 4 |- ((x e. A /\ (ph /\ -. ps)) <-> ((x e. A /\ ph) /\ -. (x e. A /\ ps)))
12	11	biabi 1181	. . 3 |- {x | (x e. A /\ (ph /\ -. ps))} = {x | ((x e. A /\ ph) /\ -. (x e. A /\ ps))}
13	1, 12	eqtr4 1122	. 2 |- ({x | (x e. A /\ ph)} \ {x | (x e. A /\ ps)}) = {x | (x e. A /\ (ph /\ -. ps))}
14		df-rab 1208	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
15		df-rab 1208	. . 3 |- {x e. A | ps} = {x | (x e. A /\ ps)}
16	14, 15	difeq12i 1586	. 2 |- ({x e. A | ph} \ {x e. A | ps}) = ({x | (x e. A /\ ph)} \ {x | (x e. A /\ ps)})
17		df-rab 1208	. 2 |- {x e. A | (ph /\ -. ps)} = {x | (x e. A /\ (ph /\ -. ps))}
18	13, 16, 17	3eqtr4 1126	1 |- ({x e. A | ph} \ {x e. A | ps}) = {x e. A | (ph /\ -. ps)}

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  {cab 1090   = wceq 1091   e. wcel 1092  {crab 1204   \ cdif 1484

This theorem is referenced by:  alephsuc3 4955

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208  df-v 1349  df-dif 1489  df-in 1491

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