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| Mirrors > Home > MPE Home > Th. List > sb8f | Structured version Visualization version GIF version | ||
| Description: Substitution of variable in universal quantifier. Version of sb8 2522 with a disjoint variable condition, not requiring ax-10 2141 or ax-13 2377. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) Avoid ax-10 2141. (Revised by SN, 5-Dec-2024.) |
| Ref | Expression |
|---|---|
| sb8f.nf | ⊢ Ⅎ𝑦𝜑 |
| Ref | Expression |
|---|---|
| sb8f | ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sb6 2085 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
| 2 | 1 | albii 1819 | . 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 ↔ ∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) |
| 3 | alcom 2159 | . 2 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑)) | |
| 4 | sb6 2085 | . . . 4 ⊢ ([𝑥 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → 𝜑)) | |
| 5 | sb8f.nf | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
| 6 | 5 | sbf 2271 | . . . 4 ⊢ ([𝑥 / 𝑦]𝜑 ↔ 𝜑) |
| 7 | equcom 2017 | . . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
| 8 | 7 | imbi1i 349 | . . . . 5 ⊢ ((𝑦 = 𝑥 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) |
| 9 | 8 | albii 1819 | . . . 4 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ ∀𝑦(𝑥 = 𝑦 → 𝜑)) |
| 10 | 4, 6, 9 | 3bitr3ri 302 | . . 3 ⊢ (∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ 𝜑) |
| 11 | 10 | albii 1819 | . 2 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥𝜑) |
| 12 | 2, 3, 11 | 3bitrri 298 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 Ⅎwnf 1783 [wsb 2064 |
| 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-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-sb 2065 |
| This theorem is referenced by: mo5f 32508 ax11-pm2 36837 bj-nfcf 36924 |
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