Theorem r19.44v 3075

Description: One direction of a restricted quantifier version of 19.44 2093. The other direction holds when 𝐴 is nonempty, see r19.44zv 4021. (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 3074	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
2		id 22	. . . 4 ⊢ (𝜓 → 𝜓)
3	2	rexlimivw 3011	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → 𝜓)
4	3	orim2i 539	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓))
5	1, 4	sylbi 206	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓))

Syntax hints:   → wi 4   ∨ wo 382  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-ral 2901  df-rex 2902

This theorem is referenced by: (None)