Theorem r2alf 2650

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

Hypothesis

Ref	Expression
r2alf.1	⊢ ℲyA

Assertion

Ref	Expression
r2alf	⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀x∀y((x ∈ A ∧ y ∈ B) → φ))

Distinct variable group:   x,y

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

Proof of Theorem r2alf

Step	Hyp	Ref	Expression
1		df-ral 2620	. 2 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀x(x ∈ A → ∀y ∈ B φ))
2		r2alf.1	. . . . . 6 ⊢ ℲyA
3	2	nfcri 2484	. . . . 5 ⊢ Ⅎy x ∈ A
4	3	19.21 1796	. . . 4 ⊢ (∀y(x ∈ A → (y ∈ B → φ)) ↔ (x ∈ A → ∀y(y ∈ B → φ)))
5		impexp 433	. . . . 5 ⊢ (((x ∈ A ∧ y ∈ B) → φ) ↔ (x ∈ A → (y ∈ B → φ)))
6	5	albii 1566	. . . 4 ⊢ (∀y((x ∈ A ∧ y ∈ B) → φ) ↔ ∀y(x ∈ A → (y ∈ B → φ)))
7		df-ral 2620	. . . . 5 ⊢ (∀y ∈ B φ ↔ ∀y(y ∈ B → φ))
8	7	imbi2i 303	. . . 4 ⊢ ((x ∈ A → ∀y ∈ B φ) ↔ (x ∈ A → ∀y(y ∈ B → φ)))
9	4, 6, 8	3bitr4i 268	. . 3 ⊢ (∀y((x ∈ A ∧ y ∈ B) → φ) ↔ (x ∈ A → ∀y ∈ B φ))
10	9	albii 1566	. 2 ⊢ (∀x∀y((x ∈ A ∧ y ∈ B) → φ) ↔ ∀x(x ∈ A → ∀y ∈ B φ))
11	1, 10	bitr4i 243	1 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀x∀y((x ∈ A ∧ y ∈ B) → φ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  Ⅎwnfc 2477  ∀wral 2615

This proof depends on 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 proof 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 used by:  r2al  2652  ralcomf  2770