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Mirrors > Home > MPE Home > Th. List > Mathboxes > mopickr | Structured version Visualization version GIF version |
Description: "At most one" picks a variable value, eliminating an existential quantifier. The proof begins with references *2.21 (pm2.21 123) and *14.26 (eupickbi 2631) from [WhiteheadRussell] p. 104 and p. 183. (Contributed by Peter Mazsa, 18-Nov-2024.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
mopickr | ⊢ ((∃*𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exancom 1863 | . . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑)) | |
2 | moeu2 37535 | . . . 4 ⊢ (∃*𝑥𝜓 ↔ (¬ ∃𝑥𝜓 ∨ ∃!𝑥𝜓)) | |
3 | 19.8a 2173 | . . . . . . . 8 ⊢ (𝜓 → ∃𝑥𝜓) | |
4 | 3 | con3i 154 | . . . . . . 7 ⊢ (¬ ∃𝑥𝜓 → ¬ 𝜓) |
5 | pm2.21 123 | . . . . . . 7 ⊢ (¬ 𝜓 → (𝜓 → 𝜑)) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ (¬ ∃𝑥𝜓 → (𝜓 → 𝜑)) |
7 | 6 | a1d 25 | . . . . 5 ⊢ (¬ ∃𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) → (𝜓 → 𝜑))) |
8 | eupickbi 2631 | . . . . . 6 ⊢ (∃!𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) ↔ ∀𝑥(𝜓 → 𝜑))) | |
9 | sp 2175 | . . . . . 6 ⊢ (∀𝑥(𝜓 → 𝜑) → (𝜓 → 𝜑)) | |
10 | 8, 9 | syl6bi 253 | . . . . 5 ⊢ (∃!𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) → (𝜓 → 𝜑))) |
11 | 7, 10 | jaoi 854 | . . . 4 ⊢ ((¬ ∃𝑥𝜓 ∨ ∃!𝑥𝜓) → (∃𝑥(𝜓 ∧ 𝜑) → (𝜓 → 𝜑))) |
12 | 2, 11 | sylbi 216 | . . 3 ⊢ (∃*𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) → (𝜓 → 𝜑))) |
13 | 1, 12 | biimtrid 241 | . 2 ⊢ (∃*𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) → (𝜓 → 𝜑))) |
14 | 13 | imp 406 | 1 ⊢ ((∃*𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 844 ∀wal 1538 ∃wex 1780 ∃*wmo 2531 ∃!weu 2561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-10 2136 ax-11 2153 ax-12 2170 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-mo 2533 df-eu 2562 |
This theorem is referenced by: (None) |
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