Theorem tfis2 4641

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 1552	. 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 4640	1 ⊢ (𝑥 ∈ On → 𝜑)

Syntax hints:   → wi 4   ↔ wb 105   ∈ wcel 2177  ∀wral 2485  Oncon0 4418

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-setind 4593

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-in 3176  df-ss 3183  df-uni 3857  df-tr 4151  df-iord 4421  df-on 4423

This theorem is referenced by:  tfis3  4642  tfrlem1  6407  ordiso2  7152  exmidontriimlem4  7352  exmidontriim  7353