Theorem difrab 2273

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 2269	. . 3 |- ({x | (x e. A /\ ph)} \ {x | (x e. A /\ ps)}) = {x | ((x e. A /\ ph) /\ -. (x e. A /\ ps))}
2		anass 439	. . . . 5 |- (((x e. A /\ ph) /\ -. ps) <-> (x e. A /\ (ph /\ -. ps)))
3		pm3.27 323	. . . . . . . 8 |- ((x e. A /\ ps) -> ps)
4	3	con3i 98	. . . . . . 7 |- (-. ps -> -. (x e. A /\ ps))
5	4	anim2i 335	. . . . . 6 |- (((x e. A /\ ph) /\ -. ps) -> ((x e. A /\ ph) /\ -. (x e. A /\ ps)))
6		pm3.2 283	. . . . . . . . 9 |- (x e. A -> (ps -> (x e. A /\ ps)))
7	6	adantr 389	. . . . . . . 8 |- ((x e. A /\ ph) -> (ps -> (x e. A /\ ps)))
8	7	con3d 95	. . . . . . 7 |- ((x e. A /\ ph) -> (-. (x e. A /\ ps) -> -. ps))
9	8	imdistani 443	. . . . . 6 |- (((x e. A /\ ph) /\ -. (x e. A /\ ps)) -> ((x e. A /\ ph) /\ -. ps))
10	5, 9	impbi 157	. . . . 5 |- (((x e. A /\ ph) /\ -. ps) <-> ((x e. A /\ ph) /\ -. (x e. A /\ ps)))
11	2, 10	bitr3 175	. . . 4 |- ((x e. A /\ (ph /\ -. ps)) <-> ((x e. A /\ ph) /\ -. (x e. A /\ ps)))
12	11	abbii 1575	. . 3 |- {x | (x e. A /\ (ph /\ -. ps))} = {x | ((x e. A /\ ph) /\ -. (x e. A /\ ps))}
13	1, 12	eqtr4 1498	. 2 |- ({x | (x e. A /\ ph)} \ {x | (x e. A /\ ps)}) = {x | (x e. A /\ (ph /\ -. ps))}
14		df-rab 1652	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
15		df-rab 1652	. . 3 |- {x e. A | ps} = {x | (x e. A /\ ps)}
16	14, 15	difeq12i 2157	. 2 |- ({x e. A | ph} \ {x e. A | ps}) = ({x | (x e. A /\ ph)} \ {x | (x e. A /\ ps)})
17		df-rab 1652	. 2 |- {x e. A | (ph /\ -. ps)} = {x | (x e. A /\ (ph /\ -. ps))}
18	13, 16, 17	3eqtr4 1505	1 |- ({x e. A | ph} \ {x e. A | ps}) = {x e. A | (ph /\ -. ps)}

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  {crab 1648   \ cdif 2044

This theorem is referenced by:  alephsuc3 7585

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652  df-v 1812  df-dif 2049  df-in 2051

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