Theorem 19.23vv

Description: Theorem 19.23v extended to two variables. (Contributed by NM, 10-Aug-2004)

	Ref	Expression
Assertion	19.23vv	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.23v	⊢ ( ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑦 𝜑 → 𝜓 ) )
2	1	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( ∃ 𝑦 𝜑 → 𝜓 ) )
3		19.23v	⊢ ( ∀ 𝑥 ( ∃ 𝑦 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 → 𝜓 ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 → 𝜓 ) )