Theorem tfis2 4727

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 1581	. 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 4726	1 ⊢ (𝑥 ∈ On → 𝜑)

Syntax hints:   → wi 4   ↔ wb 105   ∈ wcel 2209  ∀wral 2528  Oncon0 4503

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-in 3226  df-ss 3233  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508

This theorem is referenced by:  tfis3  4728  tfrlem1  6569  ordiso2  7365  exmidontriimlem4  7570  exmidontriim  7571