Theorem 2exim

Description: Theorem *11.34 in WhiteheadRussell p. 162. Theorem 19.22 of Margaris p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	2exim	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 ∃ 𝑦 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		exim	⊢ ( ∀ 𝑦 ( 𝜑 → 𝜓 ) → ( ∃ 𝑦 𝜑 → ∃ 𝑦 𝜓 ) )
2	1	aleximi	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 ∃ 𝑦 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜓 ) )