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Theorem frpoins2g 6368
Description: Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 24-Aug-2022.)
Hypotheses
Ref Expression
frpoins2g.1 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
frpoins2g.3 (𝑦 = 𝑧 → (𝜑𝜓))
Assertion
Ref Expression
frpoins2g ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝑦,𝑅,𝑧   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑧)

Proof of Theorem frpoins2g
StepHypRef Expression
1 frpoins2g.1 . 2 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
2 nfv 1912 . 2 𝑦𝜓
3 frpoins2g.3 . 2 (𝑦 = 𝑧 → (𝜑𝜓))
41, 2, 3frpoins2fg 6367 1 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086  wcel 2106  wral 3059   Po wpo 5595   Fr wfr 5638   Se wse 5639  Predcpred 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-po 5597  df-fr 5641  df-se 5642  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323
This theorem is referenced by:  frpoins3g  6369  frpoins3xpg  8164  frpoins3xp3g  8165  fpr3g  8309
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