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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