Metamath Proof Explorer
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 ) |