Theorem r19.30 3131

Description: Restricted quantifier version of 19.30 1910. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 5-Nov-2024.)

Assertion

Ref	Expression
r19.30	⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Proof of Theorem r19.30

Step	Hyp	Ref	Expression
1		pm2.53 864	. . . 4 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
2	1	ralimi 3101	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → ∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓))
3		rexnal 3116	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 𝜑)
4	3	biimpri 231	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 ¬ 𝜑)
5		rexim 3105	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
6	2, 4, 5	syl2im 41	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (¬ ∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
7	6	orrd 876	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 860  ∀wral 3078  ∃wrex 3088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1809  df-ral 3079  df-rex 3089

This theorem is used by:  disjunsn  32950  esumcvg  34485