Theorem 19.29x

Description: Variation of 19.29 with mixed quantification. (Contributed by NM, 11-Feb-2005)

	Ref	Expression
Assertion	19.29x	⊢ ( ( ∃ 𝑥 ∀ 𝑦 𝜑 ∧ ∀ 𝑥 ∃ 𝑦 𝜓 ) → ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.29r	⊢ ( ( ∃ 𝑥 ∀ 𝑦 𝜑 ∧ ∀ 𝑥 ∃ 𝑦 𝜓 ) → ∃ 𝑥 ( ∀ 𝑦 𝜑 ∧ ∃ 𝑦 𝜓 ) )
2		19.29	⊢ ( ( ∀ 𝑦 𝜑 ∧ ∃ 𝑦 𝜓 ) → ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) )
3	2	eximi	⊢ ( ∃ 𝑥 ( ∀ 𝑦 𝜑 ∧ ∃ 𝑦 𝜓 ) → ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) )
4	1 3	syl	⊢ ( ( ∃ 𝑥 ∀ 𝑦 𝜑 ∧ ∀ 𝑥 ∃ 𝑦 𝜓 ) → ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) )