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Theorem frind 4179
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 2446 . . . . . . 7 (𝜒 → ∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝜓) → 𝜑))
3 nfv 1466 . . . . . . . 8 𝑧(∀𝑦𝐴 (𝑦𝑅𝑥𝜓) → 𝜑)
4 nfv 1466 . . . . . . . . 9 𝑥𝑦𝐴 (𝑦𝑅𝑧𝜓)
5 nfs1v 1863 . . . . . . . . 9 𝑥[𝑧 / 𝑥]𝜑
64, 5nfim 1509 . . . . . . . 8 𝑥(∀𝑦𝐴 (𝑦𝑅𝑧𝜓) → [𝑧 / 𝑥]𝜑)
7 breq2 3849 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
87imbi1d 229 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑦𝑅𝑥𝜓) ↔ (𝑦𝑅𝑧𝜓)))
98ralbidv 2380 . . . . . . . . 9 (𝑥 = 𝑧 → (∀𝑦𝐴 (𝑦𝑅𝑥𝜓) ↔ ∀𝑦𝐴 (𝑦𝑅𝑧𝜓)))
10 sbequ12 1701 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
119, 10imbi12d 232 . . . . . . . 8 (𝑥 = 𝑧 → ((∀𝑦𝐴 (𝑦𝑅𝑥𝜓) → 𝜑) ↔ (∀𝑦𝐴 (𝑦𝑅𝑧𝜓) → [𝑧 / 𝑥]𝜑)))
123, 6, 11cbvral 2586 . . . . . . 7 (∀𝑥𝐴 (∀𝑦𝐴 (𝑦𝑅𝑥𝜓) → 𝜑) ↔ ∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝜓) → [𝑧 / 𝑥]𝜑))
132, 12sylib 120 . . . . . 6 (𝜒 → ∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝜓) → [𝑧 / 𝑥]𝜑))
14 frind.sb . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝜑𝜓))
1514elrab3 2772 . . . . . . . . . . 11 (𝑦𝐴 → (𝑦 ∈ {𝑥𝐴𝜑} ↔ 𝜓))
1615imbi2d 228 . . . . . . . . . 10 (𝑦𝐴 → ((𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) ↔ (𝑦𝑅𝑧𝜓)))
1716ralbiia 2392 . . . . . . . . 9 (∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) ↔ ∀𝑦𝐴 (𝑦𝑅𝑧𝜓))
1817a1i 9 . . . . . . . 8 (𝑧𝐴 → (∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) ↔ ∀𝑦𝐴 (𝑦𝑅𝑧𝜓)))
19 nfcv 2228 . . . . . . . . . 10 𝑥𝑧
20 nfcv 2228 . . . . . . . . . 10 𝑥𝐴
2119, 20, 5, 10elrabf 2769 . . . . . . . . 9 (𝑧 ∈ {𝑥𝐴𝜑} ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
2221baib 866 . . . . . . . 8 (𝑧𝐴 → (𝑧 ∈ {𝑥𝐴𝜑} ↔ [𝑧 / 𝑥]𝜑))
2318, 22imbi12d 232 . . . . . . 7 (𝑧𝐴 → ((∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) → 𝑧 ∈ {𝑥𝐴𝜑}) ↔ (∀𝑦𝐴 (𝑦𝑅𝑧𝜓) → [𝑧 / 𝑥]𝜑)))
2423ralbiia 2392 . . . . . 6 (∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) → 𝑧 ∈ {𝑥𝐴𝜑}) ↔ ∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝜓) → [𝑧 / 𝑥]𝜑))
2513, 24sylibr 132 . . . . 5 (𝜒 → ∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) → 𝑧 ∈ {𝑥𝐴𝜑}))
26 frind.fr . . . . . . . 8 (𝜒𝑅 Fr 𝐴)
27 df-frind 4159 . . . . . . . 8 (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
2826, 27sylib 120 . . . . . . 7 (𝜒 → ∀𝑠 FrFor 𝑅𝐴𝑠)
29 frind.a . . . . . . . 8 (𝜒𝐴𝑉)
30 rabexg 3982 . . . . . . . 8 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
31 frforeq3 4174 . . . . . . . . 9 (𝑠 = {𝑥𝐴𝜑} → ( FrFor 𝑅𝐴𝑠 ↔ FrFor 𝑅𝐴{𝑥𝐴𝜑}))
3231spcgv 2706 . . . . . . . 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 4158 . . . . . 6 ( FrFor 𝑅𝐴{𝑥𝐴𝜑} ↔ (∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) → 𝑧 ∈ {𝑥𝐴𝜑}) → 𝐴 ⊆ {𝑥𝐴𝜑}))
3634, 35sylib 120 . . . . 5 (𝜒 → (∀𝑧𝐴 (∀𝑦𝐴 (𝑦𝑅𝑧𝑦 ∈ {𝑥𝐴𝜑}) → 𝑧 ∈ {𝑥𝐴𝜑}) → 𝐴 ⊆ {𝑥𝐴𝜑}))
3725, 36mpd 13 . . . 4 (𝜒𝐴 ⊆ {𝑥𝐴𝜑})
38 ssrab 3099 . . . 4 (𝐴 ⊆ {𝑥𝐴𝜑} ↔ (𝐴𝐴 ∧ ∀𝑥𝐴 𝜑))
3937, 38sylib 120 . . 3 (𝜒 → (𝐴𝐴 ∧ ∀𝑥𝐴 𝜑))
4039simprd 112 . 2 (𝜒 → ∀𝑥𝐴 𝜑)
4140r19.21bi 2461 1 ((𝜒𝑥𝐴) → 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287  wcel 1438  [wsb 1692  wral 2359  {crab 2363  Vcvv 2619  wss 2999   class class class wbr 3845   FrFor wfrfor 4154   Fr wfr 4155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-frfor 4158  df-frind 4159
This theorem is referenced by: (None)
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