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Theorem frpoins2fg 6242
Description: Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 24-Aug-2022.)
Hypotheses
Ref Expression
frpoins2fg.1 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
frpoins2fg.2 𝑦𝜓
frpoins2fg.3 (𝑦 = 𝑧 → (𝜑𝜓))
Assertion
Ref Expression
frpoins2fg ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝑦,𝑅,𝑧
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑦,𝑧)

Proof of Theorem frpoins2fg
StepHypRef Expression
1 sbsbc 3721 . . . . 5 ([𝑧 / 𝑦]𝜑[𝑧 / 𝑦]𝜑)
2 frpoins2fg.2 . . . . . 6 𝑦𝜓
3 frpoins2fg.3 . . . . . 6 (𝑦 = 𝑧 → (𝜑𝜓))
42, 3sbiev 2309 . . . . 5 ([𝑧 / 𝑦]𝜑𝜓)
51, 4bitr3i 276 . . . 4 ([𝑧 / 𝑦]𝜑𝜓)
65ralbii 3092 . . 3 (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓)
7 frpoins2fg.1 . . . 4 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
87adantl 482 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑦𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
96, 8syl5bi 241 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑦𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
109frpoinsg 6241 1 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086  wnf 1786  [wsb 2067  wcel 2106  wral 3064  [wsbc 3717   Po wpo 5498   Fr wfr 5538   Se wse 5539  Predcpred 6196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-sbc 3718  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-br 5076  df-opab 5138  df-po 5500  df-fr 5541  df-se 5542  df-xp 5592  df-cnv 5594  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-pred 6197
This theorem is referenced by:  frpoins2g  6243  wfis2fg  6254
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