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Theorem tfis2d 47020
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 272 . . 3 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
4 r19.21v 3174 . . . 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 241 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 (𝜑𝜒) → (𝜑𝜓)))
93, 8tfis2 7785 . 2 (𝑥 ∈ On → (𝜑𝜓))
109com12 32 1 (𝜑 → (𝑥 ∈ On → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106  wral 3062  Oncon0 6315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-tr 5221  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-ord 6318  df-on 6319
This theorem is referenced by: (None)
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