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Mirrors > Home > ILE Home > Th. List > chvarvv | GIF version |
Description: Version of chvarv 1909 with a disjoint variable condition. (Contributed by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
chvarvv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
chvarvv.2 | ⊢ 𝜑 |
Ref | Expression |
---|---|
chvarvv | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chvarvv.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | spvv 1879 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
3 | chvarvv.2 | . 2 ⊢ 𝜑 | |
4 | 2, 3 | mpg 1427 | 1 ⊢ 𝜓 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 |
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 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1437 |
This theorem is referenced by: prodfdivap 11319 |
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