Theorem axc11n

Description: Derive set.mm's original ax-c11n from others. 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). If a disjoint variable condition is added on x and y , then this becomes an instance of aevlem . Use aecom instead when this does not lengthen the proof. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 10-May-1993) (Revised by NM, 7-Nov-2015) (Proof shortened by Wolf Lammen, 6-Mar-2018) (Revised by Wolf Lammen, 30-Nov-2019) (Proof shortened by BJ, 29-Mar-2021) (Proof shortened by Wolf Lammen, 2-Jul-2021) (New usage is discouraged.)

	Ref	Expression
Assertion	axc11n	|- ( A. x x = y -> A. y y = x )

Proof

Step	Hyp	Ref	Expression
1		dveeq1	|- ( -. A. y y = x -> ( x = z -> A. y x = z ) )
2	1	com12	|- ( x = z -> ( -. A. y y = x -> A. y x = z ) )
3		axc11r	|- ( A. x x = y -> ( A. y x = z -> A. x x = z ) )
4		aev	|- ( A. x x = z -> A. y y = x )
5	3 4	syl6	|- ( A. x x = y -> ( A. y x = z -> A. y y = x ) )
6	2 5	syl9	|- ( x = z -> ( A. x x = y -> ( -. A. y y = x -> A. y y = x ) ) )
7		ax6evr	|- E. z x = z
8	6 7	exlimiiv	|- ( A. x x = y -> ( -. A. y y = x -> A. y y = x ) )
9	8	pm2.18d	|- ( A. x x = y -> A. y y = x )