Theorem r19.30OLD 3269

Description: Obsolete version of 19.30 1884 as of 5-Nov-2024. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem r19.30OLD

Step	Hyp	Ref	Expression
1		pm2.53 848	. . . 4 ⊢ ((𝜓 ∨ 𝜑) → (¬ 𝜓 → 𝜑))
2	1	orcoms 869	. . 3 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜓 → 𝜑))
3	2	ralimi 3087	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → ∀𝑥 ∈ 𝐴 (¬ 𝜓 → 𝜑))
4		ralim 3083	. 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜓 → 𝜑) → (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑))
5		ralnex 3167	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓)
6	5	biimpri 227	. . . . 5 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜓)
7	6	imim1i 63	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑) → (¬ ∃𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜑))
8	7	orrd 860	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑) → (∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑))
9	8	orcomd 868	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
10	3, 4, 9	3syl 18	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: (None)