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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 → ( 𝜑 ↔ 𝜓 ) )
tfis2.2 ⊢ ( 𝑥 ∈ On → ( ∀ 𝑦 ∈ 𝑥 𝜓 → 𝜑 ) )
Assertion tfis2 ( 𝑥 ∈ On → 𝜑 )

Proof

Step Hyp Ref Expression
1 tfis2.1 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2 tfis2.2 ⊢ ( 𝑥 ∈ On → ( ∀ 𝑦 ∈ 𝑥 𝜓 → 𝜑 ) )
3 nfv ⊢ Ⅎ 𝑥 𝜓
4 3 1 2 tfis2f ⊢ ( 𝑥 ∈ On → 𝜑 )