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Metamath Proof Explorer

Theorem sbie

Description: Conversion of implicit substitution to explicit substitution. For versions requiring disjoint variables, but fewer axioms, see sbiev and sbievw . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 30-Jun-1994) (Revised by Mario Carneiro, 4-Oct-2016) (Proof shortened by Wolf Lammen, 13-Jul-2019) (New usage is discouraged.)

Ref Expression
Hypotheses sbie.1 ⊢ Ⅎ x ψ
sbie.2 ⊢ x = y → φ ↔ ψ
Assertion sbie ⊢ y x φ ↔ ψ

Proof

Step Hyp Ref Expression
1 sbie.1 ⊢ Ⅎ x ψ
2 sbie.2 ⊢ x = y → φ ↔ ψ
3 equsb1 ⊢ y x x = y
4 2 sbimi ⊢ y x x = y → y x φ ↔ ψ
5 3 4 ax-mp ⊢ y x φ ↔ ψ
6 1 sbf ⊢ y x ψ ↔ ψ
7 6 sblbis ⊢ y x φ ↔ ψ ↔ y x φ ↔ ψ
8 5 7 mpbi ⊢ y x φ ↔ ψ