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| Mirrors > Home > MPE Home > Th. List > axc11rv | Structured version Visualization version GIF version | ||
| Description: Version of axc11r 2371 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2007 }, from ax12v 2178 (contrary to axc11 2435 which seems to require the full ax-12 2177 and ax-13 2377, and to axc11r 2371 which seems to require the full ax-12 2177). (Contributed by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.) |
| Ref | Expression |
|---|---|
| axc11rv | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | axc16 2261 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | |
| 2 | 1 | spsd 2187 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑)) |
| 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-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 |
| This theorem is referenced by: dral1v 2372 dral1vOLD 2373 bj-axc11v 36810 |
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