MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rspc3v Structured version   Visualization version   GIF version

Theorem rspc3v 3638
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 3176 . . . 4 (𝑥 = 𝐴 → (∀𝑧𝑇 𝜑 ↔ ∀𝑧𝑇 𝜒))
3 rspc3v.2 . . . . 5 (𝑦 = 𝐵 → (𝜒𝜃))
43ralbidv 3176 . . . 4 (𝑦 = 𝐵 → (∀𝑧𝑇 𝜒 ↔ ∀𝑧𝑇 𝜃))
52, 4rspc2v 3633 . . 3 ((𝐴𝑅𝐵𝑆) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑 → ∀𝑧𝑇 𝜃))
6 rspc3v.3 . . . 4 (𝑧 = 𝐶 → (𝜃𝜓))
76rspcv 3618 . . 3 (𝐶𝑇 → (∀𝑧𝑇 𝜃𝜓))
85, 7sylan9 507 . 2 (((𝐴𝑅𝐵𝑆) ∧ 𝐶𝑇) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑𝜓))
983impa 1109 1 ((𝐴𝑅𝐵𝑆𝐶𝑇) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060
This theorem is referenced by:  rspc3dv  3641  rspc4v  3642  pocl  5605  swopolem  5607  isopolem  7365  caovassg  7631  caovcang  7634  caovordig  7638  caovordg  7640  caovdig  7647  caovdirg  7650  caofass  7736  caoftrn  7737  frpoins3xp3g  8165  prslem  18355  posi  18375  latdisdlem  18554  dlatmjdi  18581  sgrpass  18751  gaass  19328  rngdi  20178  rngdir  20179  o2timesd  20228  rglcom4d  20229  islmodd  20881  rmodislmodlem  20944  rmodislmod  20945  rmodislmodOLD  20946  lsscl  20958  assalem  21895  psmettri2  24335  xmettri2  24366  addsproplem1  28017  addsprop  28024  axtgcgrid  28486  axtg5seg  28488  axtgpasch  28490  axtgupdim2  28494  axtgeucl  28495  tgdim01  28530  f1otrgitv  28893  grpoass  30532  vcdi  30594  vcdir  30595  vcass  30596  lnolin  30783  lnopl  31943  lnfnl  31960  omndadd  33066  axtgupdim2ALTV  34662  rngodi  37891  rngodir  37892  rngoass  37893  lfli  39043  cvlexch1  39310  isthincd2lem2  48836
  Copyright terms: Public domain W3C validator