Theorem r19.44v

Description: One direction of a restricted quantifier version of 19.44 . The other direction holds when A is nonempty, see r19.44zv . (Contributed by NM, 2-Apr-2004)

	Ref	Expression
Assertion	r19.44v	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) → ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.43	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )
2		id	⊢ ( 𝜓 → 𝜓 )
3	2	rexlimivw	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜓 )
4	3	orim2i	⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) → ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ 𝜓 ) )
5	1 4	sylbi	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) → ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ 𝜓 ) )