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Theorem rspc3v 3356
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 3015 . . . 4 (𝑥 = 𝐴 → (∀𝑧𝑇 𝜑 ↔ ∀𝑧𝑇 𝜒))
3 rspc3v.2 . . . . 5 (𝑦 = 𝐵 → (𝜒𝜃))
43ralbidv 3015 . . . 4 (𝑦 = 𝐵 → (∀𝑧𝑇 𝜒 ↔ ∀𝑧𝑇 𝜃))
52, 4rspc2v 3353 . . 3 ((𝐴𝑅𝐵𝑆) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑 → ∀𝑧𝑇 𝜃))
6 rspc3v.3 . . . 4 (𝑧 = 𝐶 → (𝜃𝜓))
76rspcv 3336 . . 3 (𝐶𝑇 → (∀𝑧𝑇 𝜃𝜓))
85, 7sylan9 690 . 2 (((𝐴𝑅𝐵𝑆) ∧ 𝐶𝑇) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑𝜓))
983impa 1278 1 ((𝐴𝑅𝐵𝑆𝐶𝑇) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233
This theorem is referenced by:  swopolem  5073  isopolem  6635  caovassg  6874  caovcang  6877  caovordig  6881  caovordg  6883  caovdig  6890  caovdirg  6893  caofass  6973  caoftrn  6974  prslem  16978  posi  16997  latdisdlem  17236  dlatmjdi  17241  sgrpass  17337  gaass  17776  islmodd  18917  rmodislmodlem  18978  rmodislmod  18979  lsscl  18991  assalem  19364  psmettri2  22161  xmettri2  22192  axtgcgrid  25407  axtg5seg  25409  axtgpasch  25411  axtgupdim2  25415  axtgeucl  25416  tgdim01  25447  f1otrgitv  25795  grpoass  27485  vcdi  27548  vcdir  27549  vcass  27550  lnolin  27737  lnopl  28901  lnfnl  28918  omndadd  29834  axtgupdim2OLD  30874  rngodi  33833  rngodir  33834  rngoass  33835  lfli  34666  cvlexch1  34933  rngdir  42207
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