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 𝑥 𝜓
sbie.2 ( 𝑥 = 𝑦 → ( 𝜑𝜓 ) )
Assertion sbie ( [ 𝑦 / 𝑥 ] 𝜑𝜓 )

Proof

Step Hyp Ref Expression
1 sbie.1 𝑥 𝜓
2 sbie.2 ( 𝑥 = 𝑦 → ( 𝜑𝜓 ) )
3 equsb1 [ 𝑦 / 𝑥 ] 𝑥 = 𝑦
4 2 sbimi ( [ 𝑦 / 𝑥 ] 𝑥 = 𝑦 → [ 𝑦 / 𝑥 ] ( 𝜑𝜓 ) )
5 3 4 ax-mp [ 𝑦 / 𝑥 ] ( 𝜑𝜓 )
6 1 sbf ( [ 𝑦 / 𝑥 ] 𝜓𝜓 )
7 6 sblbis ( [ 𝑦 / 𝑥 ] ( 𝜑𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑𝜓 ) )
8 5 7 mpbi ( [ 𝑦 / 𝑥 ] 𝜑𝜓 )