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Theorem rspc3ev 3357
Description: 3-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
Hypotheses
Ref Expression
rspc3v.1 (𝑥 = 𝐴 → (𝜑𝜒))
rspc3v.2 (𝑦 = 𝐵 → (𝜒𝜃))
rspc3v.3 (𝑧 = 𝐶 → (𝜃𝜓))
Assertion
Ref Expression
rspc3ev (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → ∃𝑥𝑅𝑦𝑆𝑧𝑇 𝜑)
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 rspc3ev
StepHypRef Expression
1 simpl1 1084 . 2 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → 𝐴𝑅)
2 simpl2 1085 . 2 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → 𝐵𝑆)
3 rspc3v.3 . . . 4 (𝑧 = 𝐶 → (𝜃𝜓))
43rspcev 3340 . . 3 ((𝐶𝑇𝜓) → ∃𝑧𝑇 𝜃)
543ad2antl3 1245 . 2 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → ∃𝑧𝑇 𝜃)
6 rspc3v.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜒))
76rexbidv 3081 . . 3 (𝑥 = 𝐴 → (∃𝑧𝑇 𝜑 ↔ ∃𝑧𝑇 𝜒))
8 rspc3v.2 . . . 4 (𝑦 = 𝐵 → (𝜒𝜃))
98rexbidv 3081 . . 3 (𝑦 = 𝐵 → (∃𝑧𝑇 𝜒 ↔ ∃𝑧𝑇 𝜃))
107, 9rspc2ev 3355 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∃𝑧𝑇 𝜃) → ∃𝑥𝑅𝑦𝑆𝑧𝑇 𝜑)
111, 2, 5, 10syl3anc 1366 1 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → ∃𝑥𝑅𝑦𝑆𝑧𝑇 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942
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-rex 2947  df-v 3233
This theorem is referenced by:  f1dom3el3dif  6566  wrdl3s3  13751  pmltpclem1  23263  axlowdim  25886  axeuclidlem  25887  upgr3v3e3cycl  27158  br8d  29548  tgoldbachgt  30869  br8  31772  br6  31773  3dim1lem5  35070  lplni2  35141  jm2.27  37892
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