Theorem ralralimp 47524

Description: Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.)

Assertion

Ref	Expression
ralralimp	⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (∀𝑥 ∈ 𝐴 ((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜏,𝑥

Allowed substitution hint:   𝜃(𝑥)

Proof of Theorem ralralimp

Step	Hyp	Ref	Expression
1		ornld 1061	. . . 4 ⊢ (𝜑 → (((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏))
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏))
3	2	ralimdv 3150	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (∀𝑥 ∈ 𝐴 ((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → ∀𝑥 ∈ 𝐴 𝜏))
4		rspn0 4308	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜏 → 𝜏))
5	4	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (∀𝑥 ∈ 𝐴 𝜏 → 𝜏))
6	3, 5	syld 47	1 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (∀𝑥 ∈ 𝐴 ((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ≠ wne 2932  ∀wral 3051  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-ne 2933  df-ral 3052  df-dif 3904  df-nul 4286

This theorem is referenced by: (None)