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Mirrors > Home > MPE Home > Th. List > Mathboxes > axc11n11 | Structured version Visualization version GIF version |
Description: Proof of axc11n 2429 from { ax-1 6-- ax-7 2005, axc11 2433 } . Almost identical to axc11nfromc11 38908. (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 2433 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)) | |
2 | 1 | pm2.43i 52 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦) |
3 | equcomi 2014 | . 2 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) | |
4 | 2, 3 | sylg 1820 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-12 2175 ax-13 2375 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 |
This theorem is referenced by: (None) |
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