MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wfis2f Structured version   Visualization version   GIF version

Theorem wfis2f 6061
Description: Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
Hypotheses
Ref Expression
wfis2f.1 𝑅 We 𝐴
wfis2f.2 𝑅 Se 𝐴
wfis2f.3 𝑦𝜓
wfis2f.4 (𝑦 = 𝑧 → (𝜑𝜓))
wfis2f.5 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
Assertion
Ref Expression
wfis2f (𝑦𝐴𝜑)
Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝑦,𝑅,𝑧
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑦,𝑧)

Proof of Theorem wfis2f
StepHypRef Expression
1 wfis2f.1 . . 3 𝑅 We 𝐴
2 wfis2f.2 . . 3 𝑅 Se 𝐴
3 wfis2f.3 . . . 4 𝑦𝜓
4 wfis2f.4 . . . 4 (𝑦 = 𝑧 → (𝜑𝜓))
5 wfis2f.5 . . . 4 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
63, 4, 5wfis2fg 6060 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
71, 2, 6mp2an 688 . 2 𝑦𝐴 𝜑
87rspec 3174 1 (𝑦𝐴𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wnf 1765  wcel 2081  wral 3105   Se wse 5400   We wwe 5401  Predcpred 6022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-cnv 5451  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator