Theorem r2alf 2650

Description: Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)

Hypothesis

Ref	Expression
r2alf.1	|- F/_yA

Assertion

Ref	Expression
r2alf	|- (A.x e. A A.y e. B ph <-> A.xA.y((x e. A /\ y e. B) -> ph))

Distinct variable group:   x,y

Allowed substitution hints:   ph(x,y)   A(x,y)   B(x,y)

Proof of Theorem r2alf

Step	Hyp	Ref	Expression
1		df-ral 2620	. 2 |- (A.x e. A A.y e. B ph <-> A.x(x e. A -> A.y e. B ph))
2		r2alf.1	. . . . . 6 |- F/_yA
3	2	nfcri 2484	. . . . 5 |- F/y x e. A
4	3	19.21 1796	. . . 4 |- (A.y(x e. A -> (y e. B -> ph)) <-> (x e. A -> A.y(y e. B -> ph)))
5		impexp 433	. . . . 5 |- (((x e. A /\ y e. B) -> ph) <-> (x e. A -> (y e. B -> ph)))
6	5	albii 1566	. . . 4 |- (A.y((x e. A /\ y e. B) -> ph) <-> A.y(x e. A -> (y e. B -> ph)))
7		df-ral 2620	. . . . 5 |- (A.y e. B ph <-> A.y(y e. B -> ph))
8	7	imbi2i 303	. . . 4 |- ((x e. A -> A.y e. B ph) <-> (x e. A -> A.y(y e. B -> ph)))
9	4, 6, 8	3bitr4i 268	. . 3 |- (A.y((x e. A /\ y e. B) -> ph) <-> (x e. A -> A.y e. B ph))
10	9	albii 1566	. 2 |- (A.xA.y((x e. A /\ y e. B) -> ph) <-> A.x(x e. A -> A.y e. B ph))
11	1, 10	bitr4i 243	1 |- (A.x e. A A.y e. B ph <-> A.xA.y((x e. A /\ y e. B) -> ph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   e. wcel 1710   F/_wnfc 2477  A.wral 2615

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is referenced by:  r2al  2652  ralcomf  2770