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Theorem frind 4089
 Description: Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.)
Hypotheses
Ref Expression
frind.sb (𝑥 = 𝑦 → (𝜑𝜓))
frind.ind ((𝜒𝑥𝐴) → (∀𝑦𝐴 (𝑦𝑅𝑥𝜓) → 𝜑))
frind.fr (𝜒𝑅 Fr 𝐴)
frind.a (𝜒𝐴𝑉)
Assertion
Ref Expression
frind ((𝜒𝑥𝐴) → 𝜑)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦   𝜒,𝑥   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem frind
Dummy variables 𝑠 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frind.ind . . . . . . . 8 ((𝜒𝑥𝐴) → (∀𝑦𝐴 (𝑦𝑅𝑥𝜓) → 𝜑))
21ralrimiva 2392 . . . . . . 7 (𝜒 → ∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝜓) → 𝜑))
3 nfv 1421 . . . . . . . 8 𝑧(∀𝑦𝐴 (𝑦𝑅𝑥𝜓) → 𝜑)
4 nfv 1421 . . . . . . . . 9 𝑥𝑦𝐴 (𝑦𝑅𝑧𝜓)
5 nfs1v 1815 . . . . . . . . 9 𝑥[𝑧 / 𝑥]𝜑
64, 5nfim 1464 . . . . . . . 8 𝑥(∀𝑦𝐴 (𝑦𝑅𝑧𝜓) → [𝑧 / 𝑥]𝜑)
7 breq2 3768 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
87imbi1d 220 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑦𝑅𝑥𝜓) ↔ (𝑦𝑅𝑧𝜓)))
98ralbidv 2326 . . . . . . . . 9 (𝑥 = 𝑧 → (∀𝑦𝐴 (𝑦𝑅𝑥𝜓) ↔ ∀𝑦𝐴 (𝑦𝑅𝑧𝜓)))
10 sbequ12 1654 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
119, 10imbi12d 223 . . . . . . . 8 (𝑥 = 𝑧 → ((∀𝑦𝐴 (𝑦𝑅𝑥𝜓) → 𝜑) ↔ (∀𝑦𝐴 (𝑦𝑅𝑧𝜓) → [𝑧 / 𝑥]𝜑)))
123, 6, 11cbvral 2529 . . . . . . 7 (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝜓) → 𝜑) ↔ ∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝜓) → [𝑧 / 𝑥]𝜑))
132, 12sylib 127 . . . . . 6 (𝜒 → ∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝜓) → [𝑧 / 𝑥]𝜑))
14 frind.sb . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝜑𝜓))
1514elrab3 2699 . . . . . . . . . . 11 (𝑦𝐴 → (𝑦 ∈ {𝑥𝐴𝜑} ↔ 𝜓))
1615imbi2d 219 . . . . . . . . . 10 (𝑦𝐴 → ((𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) ↔ (𝑦𝑅𝑧𝜓)))
1716ralbiia 2338 . . . . . . . . 9 (∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) ↔ ∀𝑦𝐴 (𝑦𝑅𝑧𝜓))
1817a1i 9 . . . . . . . 8 (𝑧𝐴 → (∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) ↔ ∀𝑦𝐴 (𝑦𝑅𝑧𝜓)))
19 nfcv 2178 . . . . . . . . . 10 𝑥𝑧
20 nfcv 2178 . . . . . . . . . 10 𝑥𝐴
2119, 20, 5, 10elrabf 2696 . . . . . . . . 9 (𝑧 ∈ {𝑥𝐴𝜑} ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
2221baib 828 . . . . . . . 8 (𝑧𝐴 → (𝑧 ∈ {𝑥𝐴𝜑} ↔ [𝑧 / 𝑥]𝜑))
2318, 22imbi12d 223 . . . . . . 7 (𝑧𝐴 → ((∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) → 𝑧 ∈ {𝑥𝐴𝜑}) ↔ (∀𝑦𝐴 (𝑦𝑅𝑧𝜓) → [𝑧 / 𝑥]𝜑)))
2423ralbiia 2338 . . . . . 6 (∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) → 𝑧 ∈ {𝑥𝐴𝜑}) ↔ ∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝜓) → [𝑧 / 𝑥]𝜑))
2513, 24sylibr 137 . . . . 5 (𝜒 → ∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) → 𝑧 ∈ {𝑥𝐴𝜑}))
26 frind.fr . . . . . . . 8 (𝜒𝑅 Fr 𝐴)
27 df-frind 4069 . . . . . . . 8 (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
2826, 27sylib 127 . . . . . . 7 (𝜒 → ∀𝑠 FrFor 𝑅𝐴𝑠)
29 frind.a . . . . . . . 8 (𝜒𝐴𝑉)
30 rabexg 3900 . . . . . . . 8 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
31 frforeq3 4084 . . . . . . . . 9 (𝑠 = {𝑥𝐴𝜑} → ( FrFor 𝑅𝐴𝑠 ↔ FrFor 𝑅𝐴{𝑥𝐴𝜑}))
3231spcgv 2640 . . . . . . . 8 ({𝑥𝐴𝜑} ∈ V → (∀𝑠 FrFor 𝑅𝐴𝑠 → FrFor 𝑅𝐴{𝑥𝐴𝜑}))
3329, 30, 323syl 17 . . . . . . 7 (𝜒 → (∀𝑠 FrFor 𝑅𝐴𝑠 → FrFor 𝑅𝐴{𝑥𝐴𝜑}))
3428, 33mpd 13 . . . . . 6 (𝜒 → FrFor 𝑅𝐴{𝑥𝐴𝜑})
35 df-frfor 4068 . . . . . 6 ( FrFor 𝑅𝐴{𝑥𝐴𝜑} ↔ (∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) → 𝑧 ∈ {𝑥𝐴𝜑}) → 𝐴 ⊆ {𝑥𝐴𝜑}))
3634, 35sylib 127 . . . . 5 (𝜒 → (∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) → 𝑧 ∈ {𝑥𝐴𝜑}) → 𝐴 ⊆ {𝑥𝐴𝜑}))
3725, 36mpd 13 . . . 4 (𝜒𝐴 ⊆ {𝑥𝐴𝜑})
38 ssrab 3018 . . . 4 (𝐴 ⊆ {𝑥𝐴𝜑} ↔ (𝐴𝐴 ∧ ∀𝑥𝐴 𝜑))
3937, 38sylib 127 . . 3 (𝜒 → (𝐴𝐴 ∧ ∀𝑥𝐴 𝜑))
4039simprd 107 . 2 (𝜒 → ∀𝑥𝐴 𝜑)
4140r19.21bi 2407 1 ((𝜒𝑥𝐴) → 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   ∈ wcel 1393  [wsb 1645  ∀wral 2306  {crab 2310  Vcvv 2557   ⊆ wss 2917   class class class wbr 3764   FrFor wfrfor 4064   Fr wfr 4065 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-frfor 4068  df-frind 4069 This theorem is referenced by: (None)
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