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Theorem tfis2d 43214
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 265 . . 3 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
4 r19.21v 3139 . . . 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 234 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 (𝜑𝜒) → (𝜑𝜓)))
93, 8tfis2 7288 . 2 (𝑥 ∈ On → (𝜑𝜓))
109com12 32 1 (𝜑 → (𝑥 ∈ On → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2157  wral 3087  Oncon0 5939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-tr 4944  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-ord 5942  df-on 5943
This theorem is referenced by: (None)
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