Theorem bi2ralda 1232

Description: Formula-building rule for restricted universal quantifier (deduction rule).

Hypotheses

Ref	Expression
bi2ralda.1	⊢ (φ → ∀xφ)
bi2ralda.2	⊢ (φ → ∀yφ)
bi2ralda.3	⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ))

Assertion

Ref	Expression
bi2ralda	⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ))

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

Proof of Theorem bi2ralda

Step	Hyp	Ref	Expression
1		bi2ralda.1	. 2 ⊢ (φ → ∀xφ)
2		bi2ralda.2	. . . 4 ⊢ (φ → ∀yφ)
3		ax-17 925	. . . 4 ⊢ (x ∈ A → ∀y x ∈ A)
4	2, 3	hban 704	. . 3 ⊢ ((φ ∧ x ∈ A) → ∀y(φ ∧ x ∈ A))
5		bi2ralda.3	. . . 4 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ))
6	5	anassrs 338	. . 3 ⊢ (((φ ∧ x ∈ A) ∧ y ∈ B) → (ψ ↔ χ))
7	4, 6	biralda 1213	. 2 ⊢ ((φ ∧ x ∈ A) → (∀y ∈ B ψ ↔ ∀y ∈ B χ))
8	1, 7	biralda 1213	1 ⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ))

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

This theorem is referenced by:  bi2raldva 1233

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-ral 1205

metamath.org