Theorem eximdh

Description: Deduction from Theorem 19.22 of Margaris p. 90. (Contributed by NM, 20-May-1996)

	Ref	Expression
Hypotheses	eximdh.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	eximdh.2	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
Assertion	eximdh	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		eximdh.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		eximdh.2	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
3	2	aleximi	⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) )
4	1 3	syl	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) )