Theorem 19.21vv

Description: Compare Theorem *11.3 in WhiteheadRussell p. 161. Special case of theorem 19.21 of Margaris p. 90 with two quantifiers. See 19.21v . (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	19.21vv	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜓 → 𝜑 ) ↔ ( 𝜓 → ∀ 𝑥 ∀ 𝑦 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		19.21v	⊢ ( ∀ 𝑦 ( 𝜓 → 𝜑 ) ↔ ( 𝜓 → ∀ 𝑦 𝜑 ) )
2	1	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜓 → 𝜑 ) ↔ ∀ 𝑥 ( 𝜓 → ∀ 𝑦 𝜑 ) )
3		19.21v	⊢ ( ∀ 𝑥 ( 𝜓 → ∀ 𝑦 𝜑 ) ↔ ( 𝜓 → ∀ 𝑥 ∀ 𝑦 𝜑 ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜓 → 𝜑 ) ↔ ( 𝜓 → ∀ 𝑥 ∀ 𝑦 𝜑 ) )