Theorem tfis2d 48772

Description: Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.)

Hypotheses

Ref	Expression
tfis2d.1	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
tfis2d.2	⊢ (𝜑 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜒 → 𝜓)))

Assertion

Ref	Expression
tfis2d	⊢ (𝜑 → (𝑥 ∈ On → 𝜓))

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

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

Proof of Theorem tfis2d

Step	Hyp	Ref	Expression
1		tfis2d.1	. . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
2	1	com12 32	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒)))
3	2	pm5.74d 273	. . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
4		r19.21v 3186	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦 ∈ 𝑥 𝜒))
5		tfis2d.2	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜒 → 𝜓)))
6	5	com12 32	. . . . 5 ⊢ (𝑥 ∈ On → (𝜑 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜓)))
7	6	a2d 29	. . . 4 ⊢ (𝑥 ∈ On → ((𝜑 → ∀𝑦 ∈ 𝑥 𝜒) → (𝜑 → 𝜓)))
8	4, 7	biimtrid 242	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝜑 → 𝜒) → (𝜑 → 𝜓)))
9	3, 8	tfis2 7894	. 2 ⊢ (𝑥 ∈ On → (𝜑 → 𝜓))
10	9	com12 32	1 ⊢ (𝜑 → (𝑥 ∈ On → 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2108  ∀wral 3067  Oncon0 6395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399

This theorem is referenced by: (None)