Theorem ralralimp 42884

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 1042	. . . 4 ⊢ (𝜑 → (((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏))
2	1	adantr 473	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏))
3	2	ralimdv 3128	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (∀𝑥 ∈ 𝐴 ((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → ∀𝑥 ∈ 𝐴 𝜏))
4		rspn0 4201	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜏 → 𝜏))
5	4	adantl 474	. 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 387   ∨ wo 833   ≠ wne 2967  ∀wral 3088  ∅c0 4180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-dif 3834  df-nul 4181

This theorem is referenced by: (None)