Theorem inrab2 2272

Description: Intersection with a restricted class abstraction.

Assertion

Ref	Expression
inrab2	|- ({x e. A | ph} i^i B) = {x e. (A i^i B) | ph}

Distinct variable group:   x,B

Proof of Theorem inrab2

Step	Hyp	Ref	Expression
1		inab 2268	. . 3 |- ({x | (x e. A /\ ph)} i^i {x | x e. B}) = {x | ((x e. A /\ ph) /\ x e. B)}
2		elin 2207	. . . . . 6 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
3	2	anbi1i 481	. . . . 5 |- ((x e. (A i^i B) /\ ph) <-> ((x e. A /\ x e. B) /\ ph))
4		an23 485	. . . . 5 |- (((x e. A /\ x e. B) /\ ph) <-> ((x e. A /\ ph) /\ x e. B))
5	3, 4	bitr 173	. . . 4 |- ((x e. (A i^i B) /\ ph) <-> ((x e. A /\ ph) /\ x e. B))
6	5	abbii 1575	. . 3 |- {x | (x e. (A i^i B) /\ ph)} = {x | ((x e. A /\ ph) /\ x e. B)}
7	1, 6	eqtr4 1498	. 2 |- ({x | (x e. A /\ ph)} i^i {x | x e. B}) = {x | (x e. (A i^i B) /\ ph)}
8		df-rab 1652	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
9		abid2 1580	. . . 4 |- {x | x e. B} = B
10	9	eqcomi 1479	. . 3 |- B = {x | x e. B}
11	8, 10	ineq12i 2215	. 2 |- ({x e. A | ph} i^i B) = ({x | (x e. A /\ ph)} i^i {x | x e. B})
12		df-rab 1652	. 2 |- {x e. (A i^i B) | ph} = {x | (x e. (A i^i B) /\ ph)}
13	7, 11, 12	3eqtr4 1505	1 |- ({x e. A | ph} i^i B) = {x e. (A i^i B) | ph}

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  {crab 1648   i^i cin 2046

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-in 2051

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