Theorem ax10 1710

Description: Rederivation of ax-10 1498 from original version ax-10o 1709. See Theorem ax10o 1708 for the derivation of ax-10o 1709 from ax-10 1498.

This theorem should not be referenced in any proof. Instead, use ax-10 1498 above so that uses of ax-10 1498 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

Step	Hyp	Ref	Expression
1		ax-10o 1709	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
2	1	pm2.43i 49	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3		equcomi 1697	. . 3 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
4	3	alimi 1448	. 2 ⊢ (∀𝑦 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
5	2, 4	syl 14	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Syntax hints:   → wi 4  ∀wal 1346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-5 1440  ax-gen 1442  ax-ie2 1487  ax-8 1497  ax-17 1519  ax-i9 1523  ax-10o 1709

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)