Metamath Proof Explorer


Theorem tfis2

Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994)

Ref Expression
Hypotheses tfis2.1 ( 𝑥 = 𝑦 → ( 𝜑𝜓 ) )
tfis2.2 ( 𝑥 ∈ On → ( ∀ 𝑦𝑥 𝜓𝜑 ) )
Assertion tfis2 ( 𝑥 ∈ On → 𝜑 )

Proof

Step Hyp Ref Expression
1 tfis2.1 ( 𝑥 = 𝑦 → ( 𝜑𝜓 ) )
2 tfis2.2 ( 𝑥 ∈ On → ( ∀ 𝑦𝑥 𝜓𝜑 ) )
3 nfv 𝑥 𝜓
4 3 1 2 tfis2f ( 𝑥 ∈ On → 𝜑 )