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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-hbaeb2 | Structured version Visualization version GIF version |
Description: Biconditional version of a form of hbae 2428 with commuted quantifiers, not requiring ax-11 2152. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-hbaeb2 | ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2174 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | |
2 | axc9 2379 | . . . . 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 2363 | . . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) | |
5 | axc11 2427 | . . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)) | |
6 | 5 | pm2.43i 52 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦) |
7 | axc11r 2363 | . . . . 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 183 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) |
10 | 9 | axc4i 2313 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∀𝑧 𝑥 = 𝑦) |
11 | sp 2174 | . . 3 ⊢ (∀𝑧 𝑥 = 𝑦 → 𝑥 = 𝑦) | |
12 | 11 | alimi 1811 | . 2 ⊢ (∀𝑥∀𝑧 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦) |
13 | 10, 12 | impbii 208 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1537 |
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 1911 ax-6 1969 ax-7 2009 ax-10 2135 ax-12 2169 ax-13 2369 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-ex 1780 df-nf 1784 |
This theorem is referenced by: bj-hbaeb 36000 bj-dvv 36002 |
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