Theorem reximddv

Description: Deduction from Theorem 19.22 of Margaris p. 90. (Contributed by Thierry Arnoux, 7-Dec-2016)

	Ref	Expression
Hypotheses	reximddva.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) → 𝜒 )
	reximddva.2	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
Assertion	reximddv	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		reximddva.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) → 𝜒 )
2		reximddva.2	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
3	1	expr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) )
4	3	reximdva	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → ∃ 𝑥 ∈ 𝐴 𝜒 ) )
5	2 4	mpd	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜒 )