| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > chvarfv | Structured version Visualization version GIF version | ||
| Description: Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvar 2400 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by BJ, 31-May-2019.) |
| Ref | Expression |
|---|---|
| chvarfv.nf | ⊢ Ⅎ𝑥𝜓 |
| chvarfv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
| chvarfv.2 | ⊢ 𝜑 |
| Ref | Expression |
|---|---|
| chvarfv | ⊢ 𝜓 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | chvarfv.nf | . . 3 ⊢ Ⅎ𝑥𝜓 | |
| 2 | chvarfv.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 3 | 2 | biimpd 229 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
| 4 | 1, 3 | spimfv 2239 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
| 5 | chvarfv.2 | . 2 ⊢ 𝜑 | |
| 6 | 4, 5 | mpg 1797 | 1 ⊢ 𝜓 |
| Copyright terms: Public domain | W3C validator |