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Mirrors > Home > ILE Home > Th. List > chvarfv | GIF version |
Description: Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvar 1737 with a disjoint variable condition. (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 143 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
4 | 1, 3 | spimfv 1679 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
5 | chvarfv.2 | . 2 ⊢ 𝜑 | |
6 | 4, 5 | mpg 1431 | 1 ⊢ 𝜓 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 Ⅎwnf 1440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1441 |
This theorem is referenced by: fproddivapf 11528 fprodsplitf 11529 |
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