| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > cbv3 | Structured version Visualization version GIF version | ||
| Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 38891. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker cbv3v 2337 if possible. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.) |
| 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 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 2 | 1 | nf5ri 2195 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) |
| 3 | 2 | hbal 2167 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) |
| 4 | cbv3.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
| 5 | cbv3.3 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
| 6 | 4, 5 | spim 2392 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
| 7 | 3, 6 | alrimih 1824 | 1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 Ⅎwnf 1783 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-11 2157 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: cbval 2403 cbv1 2407 cbv3h 2409 axc16i 2441 |
| Copyright terms: Public domain | W3C validator |