This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 ) )