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Theorem wfis 5926
Description: Well-Founded Induction Schema. If all elements less than a given set 𝑥 of the well-founded class 𝐴 have a property (induction hypothesis), then all elements of 𝐴 have that property. (Contributed by Scott Fenton, 29-Jan-2011.)
Hypotheses
Ref Expression
wfis.1 𝑅 We 𝐴
wfis.2 𝑅 Se 𝐴
wfis.3 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
Assertion
Ref Expression
wfis (𝑦𝐴𝜑)
Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝑦,𝑅,𝑧
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem wfis
StepHypRef Expression
1 wfis.1 . . 3 𝑅 We 𝐴
2 wfis.2 . . 3 𝑅 Se 𝐴
3 wfis.3 . . . 4 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
43wfisg 5925 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
51, 2, 4mp2an 675 . 2 𝑦𝐴 𝜑
65rspec 3118 1 (𝑦𝐴𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2158  wral 3095  [wsbc 3630   Se wse 5265   We wwe 5266  Predcpred 5889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pr 5093
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 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-reu 3102  df-rmo 3103  df-rab 3104  df-v 3392  df-sbc 3631  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-br 4841  df-opab 4903  df-po 5229  df-so 5230  df-fr 5267  df-se 5268  df-we 5269  df-xp 5314  df-cnv 5316  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890
This theorem is referenced by: (None)
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