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 ⊢ ∀ x x = y → ∀ y y = x

Proof

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