Theorem tfis2d 44777

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 275	. . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
4		r19.21v 3175	. . . 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	syl5bi 244	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝜑 → 𝜒) → (𝜑 → 𝜓)))
9	3, 8	tfis2 7565	. 2 ⊢ (𝑥 ∈ On → (𝜑 → 𝜓))
10	9	com12 32	1 ⊢ (𝜑 → (𝑥 ∈ On → 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2110  ∀wral 3138  Oncon0 6185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-tr 5165  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-ord 6188  df-on 6189

This theorem is referenced by: (None)