Metamath Proof Explorer

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 ⊢∀xx=y→∀yy=x

Proof

Step Hyp Ref Expression
1 dveeq1 ⊢¬∀yy=x→x=z→∀yx=z
2 1 com12 ⊢x=z→¬∀yy=x→∀yx=z
3 axc11r ⊢∀xx=y→∀yx=z→∀xx=z
4 aev ⊢∀xx=z→∀yy=x
5 3 4 syl6 ⊢∀xx=y→∀yx=z→∀yy=x
6 2 5 syl9 ⊢x=z→∀xx=y→¬∀yy=x→∀yy=x
7 ax6evr ⊢∃zx=z
8 6 7 exlimiiv ⊢∀xx=y→¬∀yy=x→∀yy=x
9 8 pm2.18d ⊢∀xx=y→∀yy=x