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Theorem rspc3v 2736
Description: 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
Hypotheses
Ref Expression
rspc3v.1 (𝑥 = 𝐴 → (𝜑𝜒))
rspc3v.2 (𝑦 = 𝐵 → (𝜒𝜃))
rspc3v.3 (𝑧 = 𝐶 → (𝜃𝜓))
Assertion
Ref Expression
rspc3v ((𝐴𝑅𝐵𝑆𝐶𝑇) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑𝜓))
Distinct variable groups:   𝜓,𝑧   𝜒,𝑥   𝜃,𝑦   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝑅   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦)   𝜒(𝑦,𝑧)   𝜃(𝑥,𝑧)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝑅(𝑦,𝑧)   𝑆(𝑧)

Proof of Theorem rspc3v
StepHypRef Expression
1 rspc3v.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜒))
21ralbidv 2380 . . . 4 (𝑥 = 𝐴 → (∀𝑧𝑇 𝜑 ↔ ∀𝑧𝑇 𝜒))
3 rspc3v.2 . . . . 5 (𝑦 = 𝐵 → (𝜒𝜃))
43ralbidv 2380 . . . 4 (𝑦 = 𝐵 → (∀𝑧𝑇 𝜒 ↔ ∀𝑧𝑇 𝜃))
52, 4rspc2v 2733 . . 3 ((𝐴𝑅𝐵𝑆) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑 → ∀𝑧𝑇 𝜃))
6 rspc3v.3 . . . 4 (𝑧 = 𝐶 → (𝜃𝜓))
76rspcv 2718 . . 3 (𝐶𝑇 → (∀𝑧𝑇 𝜃𝜓))
85, 7sylan9 401 . 2 (((𝐴𝑅𝐵𝑆) ∧ 𝐶𝑇) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑𝜓))
983impa 1138 1 ((𝐴𝑅𝐵𝑆𝐶𝑇) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 924   = wceq 1289  wcel 1438  wral 2359
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
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-v 2621
This theorem is referenced by:  swopolem  4123  isopolem  5583  isosolem  5585  caovassg  5785  caovcang  5788  caovordig  5792  caovordg  5794  caovdig  5801  caovdirg  5804  caoftrn  5862
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