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Theorem wfis2f 5929
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 5928 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
71, 2, 6mp2an 675 . 2 𝑦𝐴 𝜑
87rspec 3117 1 (𝑦𝐴𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wnf 1863  wcel 2156  wral 3094   Se wse 5266   We wwe 5267  Predcpred 5890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-sep 4973  ax-nul 4981  ax-pr 5094
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ne 2977  df-ral 3099  df-rex 3100  df-reu 3101  df-rmo 3102  df-rab 3103  df-v 3391  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4115  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4843  df-opab 4905  df-po 5230  df-so 5231  df-fr 5268  df-se 5269  df-we 5270  df-xp 5315  df-cnv 5317  df-dm 5319  df-rn 5320  df-res 5321  df-ima 5322  df-pred 5891
This theorem is referenced by: (None)
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