Theorem r3al 1240

Description: Triple restricted universal quantification.

Assertion

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

Distinct variable group(s):   x,y,z   y,A,z   z,B

Proof of Theorem r3al

Step	Hyp	Ref	Expression
1		df-ral 1205	. 2 ⊢ (∀x ∈ A ∀y∀z((y ∈ B ∧ z ∈ C) → φ) ↔ ∀x(x ∈ A → ∀y∀z((y ∈ B ∧ z ∈ C) → φ)))
2		r2al 1231	. . 3 ⊢ (∀y ∈ B ∀z ∈ C φ ↔ ∀y∀z((y ∈ B ∧ z ∈ C) → φ))
3	2	biral 1223	. 2 ⊢ (∀x ∈ A ∀y ∈ B ∀z ∈ C φ ↔ ∀x ∈ A ∀y∀z((y ∈ B ∧ z ∈ C) → φ))
4		3anass 585	. . . . . . . . 9 ⊢ ((x ∈ A ∧ y ∈ B ∧ z ∈ C) ↔ (x ∈ A ∧ (y ∈ B ∧ z ∈ C)))
5	4	imbi1i 161	. . . . . . . 8 ⊢ (((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) ↔ ((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) → φ))
6		impexp 276	. . . . . . . 8 ⊢ (((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) → φ) ↔ (x ∈ A → ((y ∈ B ∧ z ∈ C) → φ)))
7	5, 6	bitr 151	. . . . . . 7 ⊢ (((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) ↔ (x ∈ A → ((y ∈ B ∧ z ∈ C) → φ)))
8	7	bial 695	. . . . . 6 ⊢ (∀z((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) ↔ ∀z(x ∈ A → ((y ∈ B ∧ z ∈ C) → φ)))
9		19.21v 942	. . . . . 6 ⊢ (∀z(x ∈ A → ((y ∈ B ∧ z ∈ C) → φ)) ↔ (x ∈ A → ∀z((y ∈ B ∧ z ∈ C) → φ)))
10	8, 9	bitr 151	. . . . 5 ⊢ (∀z((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) ↔ (x ∈ A → ∀z((y ∈ B ∧ z ∈ C) → φ)))
11	10	bial 695	. . . 4 ⊢ (∀y∀z((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) ↔ ∀y(x ∈ A → ∀z((y ∈ B ∧ z ∈ C) → φ)))
12		19.21v 942	. . . 4 ⊢ (∀y(x ∈ A → ∀z((y ∈ B ∧ z ∈ C) → φ)) ↔ (x ∈ A → ∀y∀z((y ∈ B ∧ z ∈ C) → φ)))
13	11, 12	bitr 151	. . 3 ⊢ (∀y∀z((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) ↔ (x ∈ A → ∀y∀z((y ∈ B ∧ z ∈ C) → φ)))
14	13	bial 695	. 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	3bitr4 158	1 ⊢ (∀x ∈ A ∀y ∈ B ∀z ∈ C φ ↔ ∀x∀y∀z((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∧ w3a 581  ∀wal 672   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  poss 2129  pocl 2132  dfwe2 2187

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-ral 1205

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