Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-axc11r | Structured version Visualization version GIF version |
Description: Same as axc11r 2385, but using ax12 2444 instead of ax-12 2176 directly. This better reflects axiom usage in theorems dependent on it. (Contributed by NM, 25-Jul-2015.) Avoid direct use of ax-12 2176. (Revised by Wolf Lammen, 30-Mar-2024.) |
Ref | Expression |
---|---|
wl-axc11r | ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax12 2444 | . . 3 ⊢ (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑))) | |
2 | 1 | sps 2183 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑))) |
3 | pm2.27 42 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑥 → 𝜑) → 𝜑)) | |
4 | 3 | al2imi 1815 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → ∀𝑦𝜑)) |
5 | 2, 4 | syld 47 | 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 ax-10 2144 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-nf 1784 |
This theorem is referenced by: (None) |
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