Theorem tfis3 4438

Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)

Hypotheses

Ref	Expression
tfis3.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
tfis3.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
tfis3.3	⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))

Assertion

Ref	Expression
tfis3	⊢ (𝐴 ∈ On → 𝜒)

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

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

Proof of Theorem tfis3

Step	Hyp	Ref	Expression
1		tfis3.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
2		tfis3.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3		tfis3.3	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))
4	2, 3	tfis2 4437	. 2 ⊢ (𝑥 ∈ On → 𝜑)
5	1, 4	vtoclga 2707	1 ⊢ (𝐴 ∈ On → 𝜒)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1299   ∈ wcel 1448  ∀wral 2375  Oncon0 4223

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-setind 4390

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-in 3027  df-ss 3034  df-uni 3684  df-tr 3967  df-iord 4226  df-on 4228

This theorem is referenced by:  tfisi  4439  tfrlemi1  6159  tfr1onlemaccex  6175  tfrcllemaccex  6188  tfrcl  6191