Theorem rabun2 3534

Description: Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)

Assertion

Ref	Expression
rabun2	|- {x e. (A u. B) | ph} = ({x e. A | ph} u. {x e. B | ph})

Proof of Theorem rabun2

Step	Hyp	Ref	Expression
1		df-rab 2623	. 2 |- {x e. (A u. B) | ph} = {x | (x e. (A u. B) /\ ph)}
2		df-rab 2623	. . . 4 |- {x e. A | ph} = {x | (x e. A /\ ph)}
3		df-rab 2623	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
4	2, 3	uneq12i 3416	. . 3 |- ({x e. A | ph} u. {x e. B | ph}) = ({x | (x e. A /\ ph)} u. {x | (x e. B /\ ph)})
5		elun 3220	. . . . . . 7 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
6	5	anbi1i 676	. . . . . 6 |- ((x e. (A u. B) /\ ph) <-> ((x e. A \/ x e. B) /\ ph))
7		andir 838	. . . . . 6 |- (((x e. A \/ x e. B) /\ ph) <-> ((x e. A /\ ph) \/ (x e. B /\ ph)))
8	6, 7	bitri 240	. . . . 5 |- ((x e. (A u. B) /\ ph) <-> ((x e. A /\ ph) \/ (x e. B /\ ph)))
9	8	abbii 2465	. . . 4 |- {x | (x e. (A u. B) /\ ph)} = {x | ((x e. A /\ ph) \/ (x e. B /\ ph))}
10		unab 3521	. . . 4 |- ({x | (x e. A /\ ph)} u. {x | (x e. B /\ ph)}) = {x | ((x e. A /\ ph) \/ (x e. B /\ ph))}
11	9, 10	eqtr4i 2376	. . 3 |- {x | (x e. (A u. B) /\ ph)} = ({x | (x e. A /\ ph)} u. {x | (x e. B /\ ph)})
12	4, 11	eqtr4i 2376	. 2 |- ({x e. A | ph} u. {x e. B | ph}) = {x | (x e. (A u. B) /\ ph)}
13	1, 12	eqtr4i 2376	1 |- {x e. (A u. B) | ph} = ({x e. A | ph} u. {x e. B | ph})

Syntax hints:   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339  {crab 2618   u. cun 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 2478  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214

This theorem is referenced by: (None)