Theorem reximdai

Description: Deduction from Theorem 19.22 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999)

	Ref	Expression
Hypotheses	reximdai.1	⊢ Ⅎ 𝑥 𝜑
	reximdai.2	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
Assertion	reximdai	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → ∃ 𝑥 ∈ 𝐴 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		reximdai.1	⊢ Ⅎ 𝑥 𝜑
2		reximdai.2	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
3	1 2	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )
4		rexim	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) → ( ∃ 𝑥 ∈ 𝐴 𝜓 → ∃ 𝑥 ∈ 𝐴 𝜒 ) )
5	3 4	syl	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → ∃ 𝑥 ∈ 𝐴 𝜒 ) )