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Mirrors > Home > MPE Home > Th. List > 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 3170 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | |
2 | pm3.24 405 | . . . . 5 ⊢ ¬ (𝜑 ∧ ¬ 𝜑) | |
3 | 2 | bifal 1549 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝜑) ↔ ⊥) |
4 | 3 | ralbii 3165 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ⊥) |
5 | r19.3rzv 4444 | . . . 4 ⊢ (𝐴 ≠ ∅ → (⊥ ↔ ∀𝑥 ∈ 𝐴 ⊥)) | |
6 | falim 1550 | . . . 4 ⊢ (⊥ → 𝜓) | |
7 | 5, 6 | syl6bir 256 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ⊥ → 𝜓)) |
8 | 4, 7 | syl5bi 244 | . 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) → 𝜓)) |
9 | 1, 8 | syl5bir 245 | 1 ⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ⊥wfal 1545 ≠ wne 3016 ∀wral 3138 ∅c0 4291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-dif 3939 df-nul 4292 |
This theorem is referenced by: (None) |
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