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Mirrors > Home > MPE Home > Th. List > Mathboxes > ralnralall | Structured version Visualization version GIF version |
Description: A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.) |
Ref | Expression |
---|---|
ralnralall | ⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑) → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.26 3046 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | |
2 | pm3.24 922 | . . . . 5 ⊢ ¬ (𝜑 ∧ ¬ 𝜑) | |
3 | 2 | bifal 1488 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝜑) ↔ ⊥) |
4 | 3 | ralbii 2963 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ⊥) |
5 | r19.3rzv 4016 | . . . 4 ⊢ (𝐴 ≠ ∅ → (⊥ ↔ ∀𝑥 ∈ 𝐴 ⊥)) | |
6 | falim 1489 | . . . 4 ⊢ (⊥ → 𝜓) | |
7 | 5, 6 | syl6bir 243 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ⊥ → 𝜓)) |
8 | 4, 7 | syl5bi 231 | . 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) → 𝜓)) |
9 | 1, 8 | syl5bir 232 | 1 ⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ⊥wfal 1480 ≠ wne 2780 ∀wral 2896 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-v 3175 df-dif 3543 df-nul 3875 |
This theorem is referenced by: (None) |
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