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| Mirrors > Home > MPE Home > Th. List > equsexv | Structured version Visualization version GIF version | ||
| Description: An equivalence related to implicit substitution. Version of equsex 2423 with a disjoint variable condition, which does not require ax-13 2377. See equsexvw 2004 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2267. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) Avoid ax-10 2141. (Revised by GG, 18-Nov-2024.) |
| Ref | Expression |
|---|---|
| equsalv.nf | ⊢ Ⅎ𝑥𝜓 |
| equsalv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| equsexv | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equsalv.nf | . . 3 ⊢ Ⅎ𝑥𝜓 | |
| 2 | equsalv.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 3 | 2 | biimpa 476 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜓) |
| 4 | 1, 3 | exlimi 2217 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜓) |
| 5 | 1, 2 | equsalv 2267 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
| 6 | equs4v 1999 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
| 7 | 5, 6 | sylbir 235 | . 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
| 8 | 4, 7 | impbii 209 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 ∃wex 1779 Ⅎwnf 1783 |
| 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-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: equsexhv 2292 cleljustALT2 2368 sb10f 2532 dprd2d2 20064 poimirlem25 37652 |
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