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Mirrors > Home > MPE Home > Th. List > hbae | Structured version Visualization version GIF version |
Description: All variables are effectively bound in an identical variable specifier. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 21-Apr-2018.) |
Ref | Expression |
---|---|
hbae | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2172 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | |
2 | axc9 2391 | . . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | |
3 | 1, 2 | syl7 74 | . . . 4 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
4 | axc11r 2377 | . . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) | |
5 | axc11 2444 | . . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)) | |
6 | 5 | pm2.43i 52 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦) |
7 | axc11r 2377 | . . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑦 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) | |
8 | 6, 7 | syl5 34 | . . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) |
9 | 3, 4, 8 | pm2.61ii 184 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) |
10 | 9 | axc4i 2332 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∀𝑧 𝑥 = 𝑦) |
11 | ax-11 2151 | . 2 ⊢ (∀𝑥∀𝑧 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦) | |
12 | 10, 11 | syl 17 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 |
This theorem is referenced by: hbnae 2446 nfae 2447 ax6e2eq 40768 |
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