Theorem 19.26-2

Description: Theorem 19.26 with two quantifiers. (Contributed by NM, 3-Feb-2005)

	Ref	Expression
Assertion	19.26-2	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 𝜑 ∧ ∀ 𝑥 ∀ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.26	⊢ ( ∀ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑦 𝜑 ∧ ∀ 𝑦 𝜓 ) )
2	1	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ∀ 𝑥 ( ∀ 𝑦 𝜑 ∧ ∀ 𝑦 𝜓 ) )
3		19.26	⊢ ( ∀ 𝑥 ( ∀ 𝑦 𝜑 ∧ ∀ 𝑦 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 𝜑 ∧ ∀ 𝑥 ∀ 𝑦 𝜓 ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 𝜑 ∧ ∀ 𝑥 ∀ 𝑦 𝜓 ) )