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Theorem frpoins2fg 33086
Description: Founded Partial 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 3752 . . . . 5 ([𝑧 / 𝑦]𝜑[𝑧 / 𝑦]𝜑)
2 frpoins2fg.2 . . . . . 6 𝑦𝜓
3 frpoins2fg.3 . . . . . 6 (𝑦 = 𝑧 → (𝜑𝜓))
42, 3sbiev 2330 . . . . 5 ([𝑧 / 𝑦]𝜑𝜓)
51, 4bitr3i 279 . . . 4 ([𝑧 / 𝑦]𝜑𝜓)
65ralbii 3152 . . 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 484 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑦𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))
96, 8syl5bi 244 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑦𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
109frpoinsg 33085 1 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083  wnf 1784  [wsb 2069  wcel 2114  wral 3125  [wsbc 3748   Po wpo 5444   Fr wfr 5483   Se wse 5484  Predcpred 6119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5039  df-opab 5101  df-po 5446  df-fr 5486  df-se 5487  df-xp 5533  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6120
This theorem is referenced by:  frpoins2g  33087
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