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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-axc11rc11 | Structured version Visualization version GIF version |
Description: Proving axc11r 2386 from axc11 2452. The hypotheses are two instances of
axc11 2452 used in the proof here. Some systems
introduce axc11 2452 as an
axiom, see for example System S2 in
https://us.metamath.org/downloads/finiteaxiom.pdf .
By contrast, this database sees the variant axc11r 2386, directly derived from ax-12 2177, as foundational. Later axc11 2452 is proven somewhat trickily, requiring ax-10 2145 and ax-13 2390, see its proof. (Contributed by Wolf Lammen, 18-Jul-2023.) |
Ref | Expression |
---|---|
wl-axc11rc11.1 | ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥)) |
wl-axc11rc11.2 | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Ref | Expression |
---|---|
wl-axc11rc11 | ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-axc11rc11.1 | . . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥)) | |
2 | 1 | pm2.43i 52 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥) |
3 | equcomi 2024 | . . 3 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
4 | 3 | alimi 1812 | . 2 ⊢ (∀𝑥 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦) |
5 | wl-axc11rc11.2 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | |
6 | 2, 4, 5 | 3syl 18 | 1 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 |
This theorem is referenced by: (None) |
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