Theorem r19.30 3268

Description: Restricted quantifier version of 19.30 1884. (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		pm2.53 848	. . . 4 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
2	1	ralimi 3087	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → ∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓))
3		rexnal 3169	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 𝜑)
4	3	biimpri 227	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 ¬ 𝜑)
5		rexim 3172	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
6	2, 4, 5	syl2im 40	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (¬ ∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
7	6	orrd 860	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 844  ∀wral 3064  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-ral 3069  df-rex 3070

This theorem is referenced by:  disjunsn  30933  esumcvg  32054