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Mirrors > Home > ILE Home > Th. List > cbv3 | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) |
Ref | Expression |
---|---|
cbv3.1 | ⊢ Ⅎ𝑦𝜑 |
cbv3.2 | ⊢ Ⅎ𝑥𝜓 |
cbv3.3 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
cbv3 | ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbv3.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfal 1555 | . 2 ⊢ Ⅎ𝑦∀𝑥𝜑 |
3 | cbv3.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | cbv3.3 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
5 | 3, 4 | spim 1716 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
6 | 2, 5 | alrimi 1502 | 1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 Ⅎwnf 1436 |
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-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1437 |
This theorem is referenced by: cbv3h 1721 cbv1 1722 mo2n 2025 mo23 2038 setindis 13154 bdsetindis 13156 |
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