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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 3064 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | |
2 | pm3.24 926 | . . . . 5 ⊢ ¬ (𝜑 ∧ ¬ 𝜑) | |
3 | 2 | bifal 1497 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝜑) ↔ ⊥) |
4 | 3 | ralbii 2980 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ⊥) |
5 | r19.3rzv 4064 | . . . 4 ⊢ (𝐴 ≠ ∅ → (⊥ ↔ ∀𝑥 ∈ 𝐴 ⊥)) | |
6 | falim 1498 | . . . 4 ⊢ (⊥ → 𝜓) | |
7 | 5, 6 | syl6bir 244 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ⊥ → 𝜓)) |
8 | 4, 7 | syl5bi 232 | . 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜑) → 𝜓)) |
9 | 1, 8 | syl5bir 233 | 1 ⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ⊥wfal 1488 ≠ 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-fal 1489 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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