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.

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 |

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