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Mirrors > Home > MPE Home > Th. List > cbv3v2 | Structured version Visualization version GIF version |
Description: Version of cbv3 2397 with two disjoint variable conditions, which does not require ax-11 2154 nor ax-13 2372. (Contributed by BJ, 24-Jun-2019.) (Proof shortened by Wolf Lammen, 30-Aug-2021.) |
Ref | Expression |
---|---|
cbv3v2.nf | ⊢ Ⅎ𝑥𝜓 |
cbv3v2.1 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
cbv3v2 | ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbv3v2.nf | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | cbv3v2.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
3 | 1, 2 | spimfv 2232 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
4 | 3 | alrimiv 1930 | 1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-ex 1783 df-nf 1787 |
This theorem is referenced by: bj-cbv3hv2 34977 |
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