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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 2180 | . . 3 ⊢ (∀𝑢 𝑢 = 𝑦 → 𝑢 = 𝑦) | |
2 | aev 2062 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑢 → ∀𝑥 𝑥 = 𝑦) | |
3 | pm2.21 123 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑢)) | |
4 | 2, 3 | impbid2 229 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦)) |
5 | 1, 4 | anim12i 615 | . 2 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑢 = 𝑦 ∧ (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦))) |
6 | wl-aleq 34940 | . 2 ⊢ (∀𝑥 𝑢 = 𝑦 ↔ (𝑢 = 𝑦 ∧ (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦))) | |
7 | 5, 6 | sylibr 237 | 1 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 |
This theorem is referenced by: wl-ax11-lem3 34984 |
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