![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > axc11v | Structured version Visualization version GIF version |
Description: Version of axc11 2452 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2114 }, from ax12v 2223 (contrary to axc11 2452 which seems to require the full ax-12 2222 and ax-13 2391). (Contributed by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.) |
Ref | Expression |
---|---|
axc11v | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axc16g 2292 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑦𝜑)) | |
2 | 1 | spsd 2230 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-12 2222 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1881 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |