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Mirrors > Home > MPE Home > Th. List > Mathboxes > ralralimp | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ralralimp | ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (∀𝑥 ∈ 𝐴 ((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ornld 940 | . . . 4 ⊢ (𝜑 → (((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏)) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏)) |
3 | 2 | ralimdv 2963 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (∀𝑥 ∈ 𝐴 ((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → ∀𝑥 ∈ 𝐴 𝜏)) |
4 | rspn0 3934 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜏 → 𝜏)) | |
5 | 4 | adantl 482 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (∀𝑥 ∈ 𝐴 𝜏 → 𝜏)) |
6 | 3, 5 | syld 47 | 1 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (∀𝑥 ∈ 𝐴 ((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 ≠ wne 2794 ∀wral 2912 ∅c0 3915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-v 3202 df-dif 3577 df-nul 3916 |
This theorem is referenced by: (None) |
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