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 )