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Mirrors > Home > MPE Home > Th. List > Mathboxes > alimp-no-surprise | Structured version Visualization version GIF version |
Description: There is no "surprise" in a for-all with implication if there exists a value where the antecedent is true. This is one way to prevent for-all with implication from allowing anything. For a contrast, see alimp-surprise 44315. The allsome quantifier also counters this problem, see df-alsi 44323. (Contributed by David A. Wheeler, 27-Oct-2018.) |
Ref | Expression |
---|---|
alimp-no-surprise | ⊢ ¬ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.82 1016 | . . . . 5 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑) | |
2 | 1 | albii 1799 | . . . 4 ⊢ (∀𝑥((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ∀𝑥 ¬ 𝜑) |
3 | alnex 1761 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
4 | 2, 3 | sylbb 220 | . . 3 ⊢ (∀𝑥((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) → ¬ ∃𝑥𝜑) |
5 | imnan 400 | . . 3 ⊢ ((∀𝑥((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) → ¬ ∃𝑥𝜑) ↔ ¬ (∀𝑥((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥𝜑)) | |
6 | 4, 5 | mpbi 231 | . 2 ⊢ ¬ (∀𝑥((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥𝜑) |
7 | 19.26 1850 | . . . 4 ⊢ (∀𝑥((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓))) | |
8 | 7 | anbi2ci 624 | . . 3 ⊢ ((∀𝑥((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥𝜑) ↔ (∃𝑥𝜑 ∧ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓)))) |
9 | 3anass 1086 | . . 3 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓)) ↔ (∃𝑥𝜑 ∧ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓)))) | |
10 | 3anrot 1091 | . . 3 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓)) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑)) | |
11 | 8, 9, 10 | 3bitr2i 300 | . 2 ⊢ ((∀𝑥((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑)) |
12 | 6, 11 | mtbi 323 | 1 ⊢ ¬ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1078 ∀wal 1518 ∃wex 1759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 |
This theorem depends on definitions: df-bi 208 df-an 397 df-3an 1080 df-ex 1760 |
This theorem is referenced by: alsi-no-surprise 44331 |
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