MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wfis3 Structured version   Visualization version   GIF version

Theorem wfis3 5759
Description: Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
Hypotheses
Ref Expression
wfis3.1 𝑅 We 𝐴
wfis3.2 𝑅 Se 𝐴
wfis3.3 (𝑦 = 𝑧 → (𝜑𝜓))
wfis3.4 (𝑦 = 𝐵 → (𝜑𝜒))
wfis3.5 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
Assertion
Ref Expression
wfis3 (𝐵𝐴𝜒)
Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐵   𝜒,𝑦   𝜑,𝑧   𝜓,𝑦   𝑦,𝑅,𝑧
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑧)   𝜒(𝑧)   𝐵(𝑧)

Proof of Theorem wfis3
StepHypRef Expression
1 wfis3.4 . 2 (𝑦 = 𝐵 → (𝜑𝜒))
2 wfis3.1 . . 3 𝑅 We 𝐴
3 wfis3.2 . . 3 𝑅 Se 𝐴
4 wfis3.3 . . 3 (𝑦 = 𝑧 → (𝜑𝜓))
5 wfis3.5 . . 3 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
62, 3, 4, 5wfis2 5758 . 2 (𝑦𝐴𝜑)
71, 6vtoclga 3303 1 (𝐵𝐴𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wral 2941   Se wse 5100   We wwe 5101  Predcpred 5717
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-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
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-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718
This theorem is referenced by:  omsinds  7126  uzsinds  12826
  Copyright terms: Public domain W3C validator