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| Mirrors > Home > MPE Home > Th. List > exsb | Structured version Visualization version GIF version | ||
| Description: An equivalent expression for existence. One direction (exsbim 2001) needs fewer axioms. (Contributed by NM, 2-Feb-2005.) Avoid ax-13 2377. (Revised by Wolf Lammen, 16-Oct-2022.) |
| Ref | Expression |
|---|---|
| exsb | ⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfv 1914 | . 2 ⊢ Ⅎ𝑦𝜑 | |
| 2 | nfa1 2151 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → 𝜑) | |
| 3 | ax12v 2178 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
| 4 | sp 2183 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑)) | |
| 5 | 4 | com12 32 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
| 6 | 3, 5 | impbid 212 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
| 7 | 1, 2, 6 | cbvexv1 2344 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: (None) |
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