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Theorem wfis2 6169
Description: Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
Hypotheses
Ref Expression
wfis2.1 𝑅 We 𝐴
wfis2.2 𝑅 Se 𝐴
wfis2.3 (𝑦 = 𝑧 → (𝜑𝜓))
wfis2.4 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
Assertion
Ref Expression
wfis2 (𝑦𝐴𝜑)
Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝜓,𝑦   𝑦,𝑅,𝑧
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑧)

Proof of Theorem wfis2
StepHypRef Expression
1 wfis2.1 . . 3 𝑅 We 𝐴
2 wfis2.2 . . 3 𝑅 Se 𝐴
3 wfis2.3 . . . 4 (𝑦 = 𝑧 → (𝜑𝜓))
4 wfis2.4 . . . 4 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
53, 4wfis2g 6168 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
61, 2, 5mp2an 692 . 2 𝑦𝐴 𝜑
76rspec 3120 1 (𝑦𝐴𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2114  wral 3053   Se wse 5481   We wwe 5482  Predcpred 6128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129
This theorem is referenced by:  wfis3  6170
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