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Mirrors > Home > ILE Home > Th. List > ax10 | GIF version |
Description: Rederivation of ax-10 1396 from original version ax-10o 1604. See theorem
ax10o 1603 for the derivation of ax-10o 1604 from ax-10 1396.
This theorem should not be referenced in any proof. Instead, use ax-10 1396 above so that uses of ax-10 1396 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax10 | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-10o 1604 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)) | |
2 | 1 | pm2.43i 43 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦) |
3 | equcomi 1592 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) | |
4 | 3 | alimi 1344 | . 2 ⊢ (∀𝑦 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
5 | 2, 4 | syl 14 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-5 1336 ax-gen 1338 ax-ie2 1383 ax-8 1395 ax-17 1419 ax-i9 1423 ax-10o 1604 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: (None) |
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