Theorem tfis2 4499

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

Hypotheses

Ref	Expression
tfis2.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
tfis2.2	⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))

Assertion

Ref	Expression
tfis2	⊢ (𝑥 ∈ On → 𝜑)

Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem tfis2

Step	Hyp	Ref	Expression
1		nfv 1508	. 2 ⊢ Ⅎ𝑥𝜓
2		tfis2.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3		tfis2.2	. 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))
4	1, 2, 3	tfis2f 4498	1 ⊢ (𝑥 ∈ On → 𝜑)

Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 1480  ∀wral 2416  Oncon0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290

This theorem is referenced by:  tfis3  4500  tfrlem1  6205  ordiso2  6920