Theorem inrab2 2269

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 2265	. . 3 |- ({x | (x e. A /\ ph)} i^i {x | x e. B}) = {x | ((x e. A /\ ph) /\ x e. B)}
2		elin 2204	. . . . . 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 1573	. . 3 |- {x | (x e. (A i^i B) /\ ph)} = {x | ((x e. A /\ ph) /\ x e. B)}
7	1, 6	eqtr4 1496	. 2 |- ({x | (x e. A /\ ph)} i^i {x | x e. B}) = {x | (x e. (A i^i B) /\ ph)}
8		df-rab 1650	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
9		abid2 1578	. . . 4 |- {x | x e. B} = B
10	9	eqcomi 1477	. . 3 |- B = {x | x e. B}
11	8, 10	ineq12i 2212	. 2 |- ({x e. A | ph} i^i B) = ({x | (x e. A /\ ph)} i^i {x | x e. B})
12		df-rab 1650	. 2 |- {x e. (A i^i B) | ph} = {x | (x e. (A i^i B) /\ ph)}
13	7, 11, 12	3eqtr4 1503	1 |- ({x e. A | ph} i^i B) = {x e. (A i^i B) | ph}

Syntax hints:   /\ wa 223   = wceq 955   e. wcel 957  {cab 1462  {crab 1646   i^i cin 2043

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-rab 1650  df-v 1809  df-in 2048

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