Theorem r19.45v

Description: Restricted quantifier version of one direction of 19.45 . The other direction holds when A is nonempty, see r19.45zv . (Contributed by NM, 2-Apr-2004)

	Ref	Expression
Assertion	r19.45v	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.43	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )
2		id	⊢ ( 𝜑 → 𝜑 )
3	2	rexlimivw	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜑 )
4	3	orim1i	⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) → ( 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )
5	1 4	sylbi	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )