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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 2417 with a disjoint variable condition, which does not require ax-13 2371. See equsexvw 2009 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2259. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) Avoid ax-10 2138. (Revised by Gino Giotto, 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 478 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜓) |
4 | 1, 3 | exlimi 2211 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜓) |
5 | 1, 2 | equsalv 2259 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
6 | equs4v 2004 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
7 | 5, 6 | sylbir 234 | . 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
8 | 4, 7 | impbii 208 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 ∃wex 1782 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 |
This theorem is referenced by: sb5OLD 2269 equsexhv 2289 cleljustALT2 2362 sb10f 2531 dprd2d2 19830 poimirlem25 36132 |
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