Theorem r3al 2672

Description: Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.)

Assertion

Ref	Expression
r3al	⊢ (∀x ∈ A ∀y ∈ B ∀z ∈ C φ ↔ ∀x∀y∀z((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ))

Distinct variable groups:   x,y,z   y,A,z   z,B

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

Proof of Theorem r3al

Step	Hyp	Ref	Expression
1		df-ral 2620	. 2 ⊢ (∀x ∈ A ∀y∀z((y ∈ B ∧ z ∈ C) → φ) ↔ ∀x(x ∈ A → ∀y∀z((y ∈ B ∧ z ∈ C) → φ)))
2		r2al 2652	. . 3 ⊢ (∀y ∈ B ∀z ∈ C φ ↔ ∀y∀z((y ∈ B ∧ z ∈ C) → φ))
3	2	ralbii 2639	. 2 ⊢ (∀x ∈ A ∀y ∈ B ∀z ∈ C φ ↔ ∀x ∈ A ∀y∀z((y ∈ B ∧ z ∈ C) → φ))
4		3anass 938	. . . . . . . . 9 ⊢ ((x ∈ A ∧ y ∈ B ∧ z ∈ C) ↔ (x ∈ A ∧ (y ∈ B ∧ z ∈ C)))
5	4	imbi1i 315	. . . . . . . 8 ⊢ (((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) ↔ ((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) → φ))
6		impexp 433	. . . . . . . 8 ⊢ (((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) → φ) ↔ (x ∈ A → ((y ∈ B ∧ z ∈ C) → φ)))
7	5, 6	bitri 240	. . . . . . 7 ⊢ (((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) ↔ (x ∈ A → ((y ∈ B ∧ z ∈ C) → φ)))
8	7	albii 1566	. . . . . 6 ⊢ (∀z((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) ↔ ∀z(x ∈ A → ((y ∈ B ∧ z ∈ C) → φ)))
9		19.21v 1890	. . . . . 6 ⊢ (∀z(x ∈ A → ((y ∈ B ∧ z ∈ C) → φ)) ↔ (x ∈ A → ∀z((y ∈ B ∧ z ∈ C) → φ)))
10	8, 9	bitri 240	. . . . 5 ⊢ (∀z((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) ↔ (x ∈ A → ∀z((y ∈ B ∧ z ∈ C) → φ)))
11	10	albii 1566	. . . 4 ⊢ (∀y∀z((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) ↔ ∀y(x ∈ A → ∀z((y ∈ B ∧ z ∈ C) → φ)))
12		19.21v 1890	. . . 4 ⊢ (∀y(x ∈ A → ∀z((y ∈ B ∧ z ∈ C) → φ)) ↔ (x ∈ A → ∀y∀z((y ∈ B ∧ z ∈ C) → φ)))
13	11, 12	bitri 240	. . 3 ⊢ (∀y∀z((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) ↔ (x ∈ A → ∀y∀z((y ∈ B ∧ z ∈ C) → φ)))
14	13	albii 1566	. 2 ⊢ (∀x∀y∀z((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) ↔ ∀x(x ∈ A → ∀y∀z((y ∈ B ∧ z ∈ C) → φ)))
15	1, 3, 14	3bitr4i 268	1 ⊢ (∀x ∈ A ∀y ∈ B ∀z ∈ C φ ↔ ∀x∀y∀z((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   ∈ wcel 1710  ∀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-3an 936  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: (None)