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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 2467 with commuted quantifiers, not requiring ax-11 2190. (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 2207 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | |
2 | axc9 2458 | . . . . 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 2349 | . . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) | |
5 | axc11 2466 | . . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)) | |
6 | 5 | pm2.43i 52 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦) |
7 | axc11r 2349 | . . . . 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 177 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) |
10 | 9 | axc4i 2295 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∀𝑧 𝑥 = 𝑦) |
11 | sp 2207 | . . 3 ⊢ (∀𝑧 𝑥 = 𝑦 → 𝑥 = 𝑦) | |
12 | 11 | alimi 1887 | . 2 ⊢ (∀𝑥∀𝑧 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦) |
13 | 10, 12 | impbii 199 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∀wal 1629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-10 2174 ax-12 2203 ax-13 2408 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 |
This theorem is referenced by: bj-hbaeb 33141 bj-dvv 33143 |
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