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Theorem ax10 1646
 Description: Rederivation of ax-10 1437 from original version ax-10o 1645. See theorem ax10o 1644 for the derivation of ax-10o 1645 from ax-10 1437. This theorem should not be referenced in any proof. Instead, use ax-10 1437 above so that uses of ax-10 1437 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)
Assertion
Ref Expression
ax10 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Proof of Theorem ax10
StepHypRef Expression
1 ax-10o 1645 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
21pm2.43i 48 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3 equcomi 1633 . . 3 (𝑥 = 𝑦𝑦 = 𝑥)
43alimi 1385 . 2 (∀𝑦 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
52, 4syl 14 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1283 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-5 1377  ax-gen 1379  ax-ie2 1424  ax-8 1436  ax-17 1460  ax-i9 1464  ax-10o 1645 This theorem depends on definitions:  df-bi 115 This theorem is referenced by: (None)
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