Theorem tfis2d 51077

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 33	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒)))
3	2	pm5.74d 276	. . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
4		r19.21v 3191	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦 ∈ 𝑥 𝜒))
5		tfis2d.2	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜒 → 𝜓)))
6	5	com12 33	. . . . 5 ⊢ (𝑥 ∈ On → (𝜑 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜓)))
7	6	a2d 30	. . . 4 ⊢ (𝑥 ∈ On → ((𝜑 → ∀𝑦 ∈ 𝑥 𝜒) → (𝜑 → 𝜓)))
8	4, 7	biimtrid 245	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝜑 → 𝜒) → (𝜑 → 𝜓)))
9	3, 8	tfis2 7895	. 2 ⊢ (𝑥 ∈ On → (𝜑 → 𝜓))
10	9	com12 33	1 ⊢ (𝜑 → (𝑥 ∈ On → 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2147  ∀wral 3080  Oncon0 6383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-tr 5217  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-ord 6386  df-on 6387

This theorem is used by: (None)