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Mirrors > Home > MPE Home > Th. List > chvar | Structured version Visualization version GIF version |
Description: Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) |
Ref | Expression |
---|---|
chvar.1 | ⊢ Ⅎ𝑥𝜓 |
chvar.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
chvar.3 | ⊢ 𝜑 |
Ref | Expression |
---|---|
chvar | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chvar.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | chvar.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | biimpd 230 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
4 | 1, 3 | spim 2361 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
5 | chvar.3 | . 2 ⊢ 𝜑 | |
6 | 4, 5 | mpg 1779 | 1 ⊢ 𝜓 |
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