Theorem ax10 1769

Description: Rederivation of ax-10 1558 from original version ax-10o 1768. See Theorem ax10o 1767 for the derivation of ax-10o 1768 from ax-10 1558.

This theorem should not be referenced in any proof. Instead, use ax-10 1558 above so that uses of ax-10 1558 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 1768	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
2	1	pm2.43i 49	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3		equcomi 1756	. . 3 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
4	3	alimi 1508	. 2 ⊢ (∀𝑦 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
5	2, 4	syl 14	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Syntax hints:   → wi 4  ∀wal 1400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-5 1500  ax-gen 1502  ax-ie2 1547  ax-8 1557  ax-17 1579  ax-i9 1583  ax-10o 1768

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)