Metamath Proof Explorer


Theorem sbiedv

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie ). Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sbiedvw when possible. (Contributed by NM, 7-Jan-2017) (New usage is discouraged.)

Ref Expression
Hypothesis sbiedv.1 ( ( 𝜑𝑥 = 𝑦 ) → ( 𝜓𝜒 ) )
Assertion sbiedv ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓𝜒 ) )

Proof

Step Hyp Ref Expression
1 sbiedv.1 ( ( 𝜑𝑥 = 𝑦 ) → ( 𝜓𝜒 ) )
2 nfv 𝑥 𝜑
3 nfvd ( 𝜑 → Ⅎ 𝑥 𝜒 )
4 1 ex ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓𝜒 ) ) )
5 2 3 4 sbied ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓𝜒 ) )