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Theorem tfis2d 42225
 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 261 . . 3 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
4 r19.21v 2943 . . . 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, 7syl5bi 231 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 (𝜑𝜒) → (𝜑𝜓)))
93, 8tfis2 6948 . 2 (𝑥 ∈ On → (𝜑𝜓))
109com12 32 1 (𝜑 → (𝑥 ∈ On → 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977  ∀wral 2896  Oncon0 5640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644 This theorem is referenced by: (None)
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