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Mirrors > Home > MPE Home > Th. List > cbv3hv | Structured version Visualization version GIF version |
Description: Version of cbv3h 2266 with a dv condition on 𝑥, 𝑦, which does not require ax-13 2246. Was used in a proof of axc11n 2307 (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 2024 | . 2 ⊢ Ⅎ𝑦𝜑 |
3 | cbv3hv.nf2 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
4 | 3 | nf5i 2024 | . 2 ⊢ Ⅎ𝑥𝜓 |
5 | cbv3hv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
6 | 2, 4, 5 | cbv3v 2172 | 1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-11 2034 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-ex 1705 df-nf 1710 |
This theorem is referenced by: (None) |
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