Theorem tfis3 4707

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 4706	. 2 ⊢ (𝑥 ∈ On → 𝜑)
5	1, 4	vtoclga 2880	1 ⊢ (𝐴 ∈ On → 𝜒)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∀wral 2520  Oncon0 4483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-in 3216  df-ss 3223  df-uni 3914  df-tr 4208  df-iord 4486  df-on 4488

This theorem is referenced by:  tfisi  4708  tfrlemi1  6562  tfr1onlemaccex  6578  tfrcllemaccex  6591  tfrcl  6594