This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ⊢ ∀ x x = y → ∀ y y = x

Proof

Step Hyp Ref Expression
1 ax-c11 ⊢ ∀ x x = y → ∀ x x = y → ∀ y x = y
2 1 pm2.43i ⊢ ∀ x x = y → ∀ y x = y
3 equcomi1 ⊢ x = y → y = x
4 3 alimi ⊢ ∀ y x = y → ∀ y y = x
5 2 4 syl ⊢ ∀ x x = y → ∀ y y = x