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Mirrors > Home > ILE Home > Th. List > exsb | GIF version |
Description: An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) |
Ref | Expression |
---|---|
exsb | ⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1419 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | sb8eh 1735 | . 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
3 | sb6 1766 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
4 | 3 | exbii 1496 | . 2 ⊢ (∃𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) |
5 | 2, 4 | bitri 173 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 ∃wex 1381 [wsb 1645 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-sb 1646 |
This theorem is referenced by: 2exsb 1885 |
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