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Metamath Proof Explorer

Theorem sb6a

Description: Equivalence for substitution. (Contributed by NM, 2-Jun-1993) (Proof shortened by Wolf Lammen, 23-Sep-2018)

Ref Expression
Assertion sb6a ⊢ y x φ ↔ ∀ x x = y → x y φ

Proof

Step Hyp Ref Expression
1 sbcov ⊢ y x x y φ ↔ y x φ
2 sb6 ⊢ y x x y φ ↔ ∀ x x = y → x y φ
3 1 2 bitr3i ⊢ y x φ ↔ ∀ x x = y → x y φ