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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 2419 with a disjoint variable condition, which does not require ax-13 2373. See equsexvw 2016 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2268. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
equsalv.nf | ⊢ Ⅎ𝑥𝜓 |
equsalv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
equsexv | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbalex 2244 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | equsalv.nf | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | equsalv.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | equsalv 2268 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
5 | 1, 4 | bitri 278 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1540 ∃wex 1786 Ⅎwnf 1790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 |
This theorem is referenced by: sb5OLD 2277 equsexhv 2297 cleljustALT2 2366 sb10f 2533 dprd2d2 19297 poimirlem25 35457 |
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