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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 2396 with a disjoint variable condition, which does not require ax-13 2373. (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 228 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
4 | 1, 3 | spimfv 2233 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
5 | chvarfv.2 | . 2 ⊢ 𝜑 | |
6 | 4, 5 | mpg 1800 | 1 ⊢ 𝜓 |
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