Theorem tfis2d 49673

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 3159	. . . 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 7836	. 2 ⊢ (𝑥 ∈ On → (𝜑 → 𝜓))
10	9	com12 32	1 ⊢ (𝜑 → (𝑥 ∈ On → 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2109  ∀wral 3045  Oncon0 6335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-tr 5218  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-ord 6338  df-on 6339

This theorem is referenced by: (None)