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Mirrors > Home > ILE Home > Th. List > chvarvv | GIF version |
Description: Version of chvarv 1930 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 1900 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
3 | chvarvv.2 | . 2 ⊢ 𝜑 | |
4 | 2, 3 | mpg 1444 | 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 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-nf 1454 |
This theorem is referenced by: prodfdivap 11497 |
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