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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-axc11rc11 | Structured version Visualization version GIF version |
Description: Proving axc11r 2366 from axc11 2430. The hypotheses are two instances of
axc11 2430 used in the proof here. Some systems
introduce axc11 2430 as an
axiom, see for example System S2 in
https://us.metamath.org/downloads/finiteaxiom.pdf 2430.
By contrast, this database sees the variant axc11r 2366, directly derived from ax-12 2171, as foundational. Later axc11 2430 is proven somewhat trickily, requiring ax-10 2137 and ax-13 2372, 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 2020 | . . 3 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
4 | 3 | alimi 1814 | . 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 1537 |
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 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 |
This theorem is referenced by: (None) |
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