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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 1999) needs fewer axioms. (Contributed by NM, 2-Feb-2005.) Avoid ax-13 2375. (Revised by Wolf Lammen, 16-Oct-2022.) |
Ref | Expression |
---|---|
exsb | ⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1912 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfa1 2149 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → 𝜑) | |
3 | ax12v 2176 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
4 | sp 2181 | . . . 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 2343 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 ∃wex 1776 |
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 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-11 2155 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 |
This theorem is referenced by: (None) |
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