Theorem r19.45v 3201

Description: Restricted quantifier version of one direction of 19.45 2277. The other direction holds when 𝐴 is nonempty, see r19.45zv 4471. (Contributed by NM, 2-Apr-2004.)

Assertion

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

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem r19.45v

Step	Hyp	Ref	Expression
1		r19.43 3135	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
2		id 23	. . . 4 ⊢ (𝜑 → 𝜑)
3	2	rexlimivw 3164	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜑)
4	3	orim1i 923	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) → (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
5	1, 4	sylbi 220	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861  ∃wrex 3091

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 3082  df-rex 3092

This theorem is used by: (None)