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| Mirrors > Home > MPE Home > Th. List > Mathboxes > axc11n11 | Structured version Visualization version GIF version | ||
| Description: Proof of axc11n 2431 from { ax-1 6-- ax-7 2007, axc11 2435 } . Almost identical to axc11nfromc11 38927. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| axc11n11 | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | axc11 2435 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)) | |
| 2 | 1 | pm2.43i 52 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦) | 
| 3 | equcomi 2016 | . 2 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) | |
| 4 | 2, 3 | sylg 1823 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 ax-13 2377 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 | 
| This theorem is referenced by: (None) | 
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