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Theorem rspc3ev 3624
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 1189 . 2 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → 𝐴𝑅)
2 simpl2 1190 . 2 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → 𝐵𝑆)
3 rspc3v.3 . . . 4 (𝑧 = 𝐶 → (𝜃𝜓))
43rspcev 3607 . . 3 ((𝐶𝑇𝜓) → ∃𝑧𝑇 𝜃)
543ad2antl3 1185 . 2 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → ∃𝑧𝑇 𝜃)
6 rspc3v.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜒))
76rexbidv 3173 . . 3 (𝑥 = 𝐴 → (∃𝑧𝑇 𝜑 ↔ ∃𝑧𝑇 𝜒))
8 rspc3v.2 . . . 4 (𝑦 = 𝐵 → (𝜒𝜃))
98rexbidv 3173 . . 3 (𝑦 = 𝐵 → (∃𝑧𝑇 𝜒 ↔ ∃𝑧𝑇 𝜃))
107, 9rspc2ev 3620 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∃𝑧𝑇 𝜃) → ∃𝑥𝑅𝑦𝑆𝑧𝑇 𝜑)
111, 2, 5, 10syl3anc 1369 1 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → ∃𝑥𝑅𝑦𝑆𝑧𝑇 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wrex 3065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066
This theorem is referenced by:  f1dom3el3dif  7273  wrdl3s3  14937  pmltpclem1  25364  axlowdim  28759  axeuclidlem  28760  upgr3v3e3cycl  29977  br8d  32383  tgoldbachgt  34231  2goelgoanfmla1  34970  br8  35286  br6  35287  3dim1lem5  38876  lplni2  38947  3rspcedvdw  41621  3cubes  42032  jm2.27  42351
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