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Theorem wfis3 5624
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 5623 . 2 (𝑦𝐴𝜑)
71, 6vtoclga 3244 1 (𝐵𝐴𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1976  wral 2895   Se wse 4985   We wwe 4986  Predcpred 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-cnv 5036  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583
This theorem is referenced by:  omsinds  6953  uzsinds  12603
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