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Theorem wfis3 6171
 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 6170 . 2 (𝑦𝐴𝜑)
71, 6vtoclga 3494 1 (𝐵𝐴𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∀wral 3070   Se wse 5484   We wwe 5485  Predcpred 6129 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-po 5446  df-so 5447  df-fr 5486  df-se 5487  df-we 5488  df-xp 5533  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130 This theorem is referenced by:  omsinds  7604  uzsinds  13409
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