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Mirrors > Home > MPE Home > Th. List > Mathboxes > axc11n11 | Structured version Visualization version GIF version |
Description: Proof of axc11n 2425 from { ax-1 6-- ax-7 2016, axc11 2429 } . Almost identical to axc11nfromc11 36677. (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 2429 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)) | |
2 | 1 | pm2.43i 52 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦) |
3 | equcomi 2025 | . 2 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) | |
4 | 2, 3 | sylg 1830 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-12 2175 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-nf 1792 |
This theorem is referenced by: (None) |
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