Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cbv3h | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.) |
Ref | Expression |
---|---|
cbv3h.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
cbv3h.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
cbv3h.3 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
cbv3h | ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbv3h.1 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | nfi 1351 | . 2 ⊢ Ⅎ𝑦𝜑 |
3 | cbv3h.2 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
4 | 3 | nfi 1351 | . 2 ⊢ Ⅎ𝑥𝜓 |
5 | cbv3h.3 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
6 | 2, 4, 5 | cbv3 1630 | 1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: cbvalh 1636 ax16 1694 ax16i 1738 cleqh 2137 |
Copyright terms: Public domain | W3C validator |