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| Mirrors > Home > MPE Home > Th. List > cbv3hv | Structured version Visualization version GIF version | ||
| Description: Rule used to change bound variables, using implicit substitution. Version of cbv3h 2409 with a disjoint variable condition on 𝑥, 𝑦, which does not require ax-13 2377. Was used in a proof of axc11n 2431 (but of independent interest). (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Nov-2020.) (Proof shortened by BJ, 30-Nov-2020.) |
| Ref | Expression |
|---|---|
| cbv3hv.nf1 | ⊢ (𝜑 → ∀𝑦𝜑) |
| cbv3hv.nf2 | ⊢ (𝜓 → ∀𝑥𝜓) |
| cbv3hv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
| Ref | Expression |
|---|---|
| cbv3hv | ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbv3hv.nf1 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
| 2 | 1 | nf5i 2146 | . 2 ⊢ Ⅎ𝑦𝜑 |
| 3 | cbv3hv.nf2 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
| 4 | 3 | nf5i 2146 | . 2 ⊢ Ⅎ𝑥𝜓 |
| 5 | cbv3hv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
| 6 | 2, 4, 5 | cbv3v 2337 | 1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 |
| 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-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: (None) |
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