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Theorem rspc3v 3478
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 3133 . . . 4 (𝑥 = 𝐴 → (∀𝑧𝑇 𝜑 ↔ ∀𝑧𝑇 𝜒))
3 rspc3v.2 . . . . 5 (𝑦 = 𝐵 → (𝜒𝜃))
43ralbidv 3133 . . . 4 (𝑦 = 𝐵 → (∀𝑧𝑇 𝜒 ↔ ∀𝑧𝑇 𝜃))
52, 4rspc2v 3475 . . 3 ((𝐴𝑅𝐵𝑆) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑 → ∀𝑧𝑇 𝜃))
6 rspc3v.3 . . . 4 (𝑧 = 𝐶 → (𝜃𝜓))
76rspcv 3458 . . 3 (𝐶𝑇 → (∀𝑧𝑇 𝜃𝜓))
85, 7sylan9 503 . 2 (((𝐴𝑅𝐵𝑆) ∧ 𝐶𝑇) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑𝜓))
983impa 1136 1 ((𝐴𝑅𝐵𝑆𝐶𝑇) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-v 3352
This theorem is referenced by:  swopolem  5209  isopolem  6789  caovassg  7032  caovcang  7035  caovordig  7039  caovordg  7041  caovdig  7048  caovdirg  7051  caofass  7131  caoftrn  7132  prslem  17200  posi  17219  latdisdlem  17458  dlatmjdi  17463  sgrpass  17559  gaass  17996  islmodd  19141  rmodislmodlem  19202  rmodislmod  19203  lsscl  19215  assalem  19593  psmettri2  22396  xmettri2  22427  axtgcgrid  25656  axtg5seg  25658  axtgpasch  25660  axtgupdim2  25664  axtgeucl  25665  tgdim01  25696  f1otrgitv  26044  grpoass  27817  vcdi  27879  vcdir  27880  vcass  27881  lnolin  28068  lnopl  29232  lnfnl  29249  omndadd  30156  axtgupdim2OLD  31200  rngodi  34128  rngodir  34129  rngoass  34130  lfli  35020  cvlexch1  35287  rngdir  42554
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