Theorem difrab 2868

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 2863	. . 3 |- ({x | (x e. A /\ ph)} \ {x | (x e. A /\ ps)}) = {x | ((x e. A /\ ph) /\ -. (x e. A /\ ps))}
2		anass 487	. . . . 5 |- (((x e. A /\ ph) /\ -. ps) <-> (x e. A /\ (ph /\ -. ps)))
3		simpr 350	. . . . . . . 8 |- ((x e. A /\ ps) -> ps)
4	3	con3i 114	. . . . . . 7 |- (-. ps -> -. (x e. A /\ ps))
5	4	anim2i 362	. . . . . 6 |- (((x e. A /\ ph) /\ -. ps) -> ((x e. A /\ ph) /\ -. (x e. A /\ ps)))
6		pm3.2 305	. . . . . . . . 9 |- (x e. A -> (ps -> (x e. A /\ ps)))
7	6	adantr 425	. . . . . . . 8 |- ((x e. A /\ ph) -> (ps -> (x e. A /\ ps)))
8	7	con3d 111	. . . . . . 7 |- ((x e. A /\ ph) -> (-. (x e. A /\ ps) -> -. ps))
9	8	imdistani 491	. . . . . 6 |- (((x e. A /\ ph) /\ -. (x e. A /\ ps)) -> ((x e. A /\ ph) /\ -. ps))
10	5, 9	impbii 174	. . . . 5 |- (((x e. A /\ ph) /\ -. ps) <-> ((x e. A /\ ph) /\ -. (x e. A /\ ps)))
11	2, 10	bitr3i 192	. . . 4 |- ((x e. A /\ (ph /\ -. ps)) <-> ((x e. A /\ ph) /\ -. (x e. A /\ ps)))
12	11	abbii 2006	. . 3 |- {x | (x e. A /\ (ph /\ -. ps))} = {x | ((x e. A /\ ph) /\ -. (x e. A /\ ps))}
13	1, 12	eqtr4i 1911	. 2 |- ({x | (x e. A /\ ph)} \ {x | (x e. A /\ ps)}) = {x | (x e. A /\ (ph /\ -. ps))}
14		df-rab 2112	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
15		df-rab 2112	. . 3 |- {x e. A | ps} = {x | (x e. A /\ ps)}
16	14, 15	difeq12i 2724	. 2 |- ({x e. A | ph} \ {x e. A | ps}) = ({x | (x e. A /\ ph)} \ {x | (x e. A /\ ps)})
17		df-rab 2112	. 2 |- {x e. A | (ph /\ -. ps)} = {x | (x e. A /\ (ph /\ -. ps))}
18	13, 16, 17	3eqtr4i 1921	1 |- ({x e. A | ph} \ {x e. A | ps}) = {x e. A | (ph /\ -. ps)}

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  {crab 2108   \ cdif 2590

This theorem is referenced by:  alephsuc3 8854

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rab 2112  df-v 2294  df-dif 2597

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