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Theorem tfis2d 49199
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.)
Hypotheses
Ref Expression
tfis2d.1 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
tfis2d.2 (𝜑 → (𝑥 ∈ On → (∀𝑦𝑥 𝜒𝜓)))
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
StepHypRef Expression
1 tfis2d.1 . . . . 5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
21com12 32 . . . 4 (𝑥 = 𝑦 → (𝜑 → (𝜓𝜒)))
32pm5.74d 273 . . 3 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
4 r19.21v 3180 . . . 4 (∀𝑦𝑥 (𝜑𝜒) ↔ (𝜑 → ∀𝑦𝑥 𝜒))
5 tfis2d.2 . . . . . 6 (𝜑 → (𝑥 ∈ On → (∀𝑦𝑥 𝜒𝜓)))
65com12 32 . . . . 5 (𝑥 ∈ On → (𝜑 → (∀𝑦𝑥 𝜒𝜓)))
76a2d 29 . . . 4 (𝑥 ∈ On → ((𝜑 → ∀𝑦𝑥 𝜒) → (𝜑𝜓)))
84, 7biimtrid 242 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 (𝜑𝜒) → (𝜑𝜓)))
93, 8tfis2 7878 . 2 (𝑥 ∈ On → (𝜑𝜓))
109com12 32 1 (𝜑 → (𝑥 ∈ On → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  wral 3061  Oncon0 6384
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388
This theorem is referenced by: (None)
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