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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-axc11rc11 | Structured version Visualization version GIF version |
Description: Proving axc11r 2385 from axc11 2451. The hypotheses are two instances of
axc11 2451 used in the proof here. Some systems
introduce axc11 2451 as an
axiom, see for example System S2 in
https://us.metamath.org/downloads/finiteaxiom.pdf .
By contrast, this database sees the variant axc11r 2385, directly derived from ax-12 2176, as foundational. Later axc11 2451 is proven somewhat trickily, requiring ax-10 2144 and ax-13 2389, 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 2023 | . . 3 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
4 | 3 | alimi 1811 | . 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 1534 |
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 1969 ax-7 2014 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 |
This theorem is referenced by: (None) |
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