Theorem r19.44v 3203

Description: One direction of a restricted quantifier version of 19.44 2276. The other direction holds when 𝐴 is nonempty, see r19.44zv 4475. (Contributed by NM, 2-Apr-2004.)

Assertion

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

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem r19.44v

Step	Hyp	Ref	Expression
1		r19.43 3136	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
2		id 23	. . . 4 ⊢ (𝜓 → 𝜓)
3	2	rexlimivw 3165	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → 𝜓)
4	3	orim2i 924	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓))
5	1, 4	sylbi 220	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861  ∃wrex 3092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-ral 3083  df-rex 3093

This theorem is used by: (None)