Proof that Equality is Symmetric

9. Using our logical axioms and structural rules as discussed in class (but not the axioms of ZFC), give a formal proof that equality is symmetric: i.e., prove ⊢∀x∀y(x=y→y=x) in our formal system.

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Statement: For all x and y, if x = y this implies y = x. ⊢∀x∀y(x=y→y=x)

Proof:

Expression	Axiom/Explanation
x = y ⊢ x = y	Axion of Assumption
x = y ⊢ x = x	=I
(x = y) ⊢ (y = x)	=E
⊢ (x = y → y = x)	→I
⊢ ∀x ∀y (x = y → y = x)	∀I