Theorem ax10from10o 2177

Description: Rederivation of ax-10 2140 from original version ax-10o 2139. See Theorem ax10o 1952 for the derivation of ax-10o 2139 from ax-10 2140.

This theorem should not be referenced in any proof. Instead, use ax-10 2140 above so that uses of ax-10 2140 can be more easily identified, or use aecom-o 2151 when this form is needed for studies involving ax-10o 2139 and omitting ax-17 1616. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax10from10o	⊢ (∀x x = y → ∀y y = x)

Proof of Theorem ax10from10o

Step	Hyp	Ref	Expression
1		ax-10o 2139	. . 3 ⊢ (∀x x = y → (∀x x = y → ∀y x = y))
2	1	pm2.43i 43	. 2 ⊢ (∀x x = y → ∀y x = y)
3		equcomi 1679	. . 3 ⊢ (x = y → y = x)
4	3	alimi 1559	. 2 ⊢ (∀y x = y → ∀y y = x)
5	2, 4	syl 15	1 ⊢ (∀x x = y → ∀y y = x)

Syntax hints:   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-10o 2139

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by: (None)