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 36041. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker cbv3v 2355 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 2174 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) |
4 | cbv3.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
5 | cbv3.3 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
6 | 4, 5 | spim 2405 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
7 | 3, 6 | alrimih 1824 | 1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 |
This theorem is referenced by: cbval 2416 cbv1 2422 cbv3h 2424 axc16i 2458 |
Copyright terms: Public domain | W3C validator |