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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-hbnaev | Structured version Visualization version GIF version |
Description: Any variable is free in ¬ ∀𝑥𝑥 = 𝑦, if 𝑥 and 𝑦 are distinct. The latter condition can actually be lifted, but this version is easier to prove. The proof does not use ax-10 2006. (Contributed by Wolf Lammen, 9-Apr-2021.) |
Ref | Expression |
---|---|
wl-hbnaev | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aev 1970 | . . 3 ⊢ (∀𝑢 𝑢 = 𝑡 → ∀𝑥 𝑥 = 𝑦) | |
2 | 1 | con3i 149 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑡) |
3 | ax-5 1827 | . 2 ⊢ (¬ ∀𝑢 𝑢 = 𝑡 → ∀𝑧 ¬ ∀𝑢 𝑢 = 𝑡) | |
4 | aev 1970 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑡) | |
5 | 4 | con3i 149 | . . 3 ⊢ (¬ ∀𝑢 𝑢 = 𝑡 → ¬ ∀𝑥 𝑥 = 𝑦) |
6 | 5 | alimi 1730 | . 2 ⊢ (∀𝑧 ¬ ∀𝑢 𝑢 = 𝑡 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦) |
7 | 2, 3, 6 | 3syl 18 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: (None) |
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