Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ax13v | Structured version Visualization version GIF version |
Description: A weaker version of ax-13 2302 with distinct variable restrictions on pairs
𝑥,
𝑧 and 𝑦, 𝑧. In order to show (with
ax13 2305) that this
weakening is still adequate, this should be the only theorem referencing
ax-13 2302 directly.
Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1870. So essentially this theorem states, that a distinct variable condition can be replaced with an inequality between set variables. (Contributed by NM, 30-Jun-2016.) |
Ref | Expression |
---|---|
ax13v | ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-13 2302 | 1 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1506 |
This theorem was proved from axioms: ax-13 2302 |
This theorem is referenced by: ax13lem1 2304 wl-spae 34244 |
Copyright terms: Public domain | W3C validator |