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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-ax11-lem2 | Structured version Visualization version GIF version |
Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) |
Ref | Expression |
---|---|
wl-ax11-lem2 | ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2206 | . . 3 ⊢ (∀𝑢 𝑢 = 𝑦 → 𝑢 = 𝑦) | |
2 | aev 2139 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑢 → ∀𝑥 𝑥 = 𝑦) | |
3 | pm2.21 121 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑢)) | |
4 | 2, 3 | impbid2 216 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦)) |
5 | 1, 4 | anim12i 592 | . 2 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑢 = 𝑦 ∧ (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦))) |
6 | wl-aleq 33650 | . 2 ⊢ (∀𝑥 𝑢 = 𝑦 ↔ (𝑢 = 𝑦 ∧ (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦))) | |
7 | 5, 6 | sylibr 224 | 1 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 ∀wal 1628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-10 2173 ax-12 2202 ax-13 2407 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-tru 1633 df-ex 1852 df-nf 1857 |
This theorem is referenced by: wl-ax11-lem3 33691 |
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