Theorem aecom-o 2151

Description: Commutation law for identical variable specifiers. The antecedent and consequent are true when x and y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 1946 using ax-10o 2139. Unlike ax10from10o 2177, this version does not require ax-17 1616. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
aecom-o	⊢ (∀x x = y → ∀y y = x)

Proof of Theorem aecom-o

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:  aecoms-o  2152  naecoms-o  2178  aev-o  2182  ax11indalem  2197