Theorem aecom-o

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 using ax-c11 . Unlike axc11nfromc11 , this version does not require ax-5 (see comment of equcomi1 ). (Contributed by NM, 10-May-1993) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	aecom-o	$$⊢\forall xx=y\to \forall yy=x$$

Proof

Step	Hyp	Ref	Expression
1		ax-c11	$$⊢\forall xx=y\to \left(\forall xx=y\to \forall yx=y\right)$$
2	1	pm2.43i	$$⊢\forall xx=y\to \forall yx=y$$
3		equcomi1	$$⊢x=y\to y=x$$
4	3	alimi	$$⊢\forall yx=y\to \forall yy=x$$
5	2 4	syl	$$⊢\forall xx=y\to \forall yy=x$$