Users' Mathboxes Mathbox for Emmett Weisz < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tfis2d Structured version   Visualization version   GIF version

Theorem tfis2d 42752
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 262 . . 3 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
4 r19.21v 2989 . . . 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 232 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 (𝜑𝜒) → (𝜑𝜓)))
93, 8tfis2 7098 . 2 (𝑥 ∈ On → (𝜑𝜓))
109com12 32 1 (𝜑 → (𝑥 ∈ On → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2030  wral 2941  Oncon0 5761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator