Theorem reximdd

Description: Deduction from Theorem 19.22 of Margaris p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022)

	Ref	Expression
Hypotheses	reximdd.1	⊢ Ⅎ 𝑥 𝜑
	reximdd.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓 ) → 𝜒 )
	reximdd.3	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
Assertion	reximdd	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		reximdd.1	⊢ Ⅎ 𝑥 𝜑
2		reximdd.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓 ) → 𝜒 )
3		reximdd.3	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
4	2	3exp	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
5	1 4	reximdai	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → ∃ 𝑥 ∈ 𝐴 𝜒 ) )
6	3 5	mpd	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜒 )