Theorem 19.28vv

Description: Theorem *11.47 in WhiteheadRussell p. 164. Theorem 19.28 of Margaris p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	19.28vv	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜓 ∧ 𝜑 ) ↔ ( 𝜓 ∧ ∀ 𝑥 ∀ 𝑦 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		19.28v	⊢ ( ∀ 𝑦 ( 𝜓 ∧ 𝜑 ) ↔ ( 𝜓 ∧ ∀ 𝑦 𝜑 ) )
2	1	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜓 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝜓 ∧ ∀ 𝑦 𝜑 ) )
3		19.28v	⊢ ( ∀ 𝑥 ( 𝜓 ∧ ∀ 𝑦 𝜑 ) ↔ ( 𝜓 ∧ ∀ 𝑥 ∀ 𝑦 𝜑 ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜓 ∧ 𝜑 ) ↔ ( 𝜓 ∧ ∀ 𝑥 ∀ 𝑦 𝜑 ) )