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Metamath Proof Explorer

Theorem 19.21-2

Description: Version of 19.21 with two quantifiers. (Contributed by NM, 4-Feb-2005)

Ref Expression
Hypotheses 19.21-2.1 ⊢ Ⅎ x φ
19.21-2.2 ⊢ Ⅎ y φ
Assertion 19.21-2 ⊢ ∀ x ∀ y φ → ψ ↔ φ → ∀ x ∀ y ψ

Proof

Step Hyp Ref Expression
1 19.21-2.1 ⊢ Ⅎ x φ
2 19.21-2.2 ⊢ Ⅎ y φ
3 2 19.21 ⊢ ∀ y φ → ψ ↔ φ → ∀ y ψ
4 3 albii ⊢ ∀ x ∀ y φ → ψ ↔ ∀ x φ → ∀ y ψ
5 1 19.21 ⊢ ∀ x φ → ∀ y ψ ↔ φ → ∀ x ∀ y ψ
6 4 5 bitri ⊢ ∀ x ∀ y φ → ψ ↔ φ → ∀ x ∀ y ψ