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Mirrors > Home > MPE Home > Th. List > cbv3v | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Version of cbv3 2392 with a disjoint variable condition, which does not require ax-13 2367. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
cbv3v.nf1 | ⊢ Ⅎ𝑦𝜑 |
cbv3v.nf2 | ⊢ Ⅎ𝑥𝜓 |
cbv3v.1 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
cbv3v | ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbv3v.nf1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nf5ri 2184 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | hbal 2157 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) |
4 | cbv3v.nf2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
5 | cbv3v.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
6 | 4, 5 | spimfv 2228 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
7 | 3, 6 | alrimih 1819 | 1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1532 Ⅎwnf 1778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-11 2147 ax-12 2167 |
This theorem depends on definitions: df-bi 206 df-ex 1775 df-nf 1779 |
This theorem is referenced by: cbv1v 2328 cbv3hv 2332 cbvalv1 2333 |
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