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Theorem rspc3ev 2686
Description: 3-variable restricted existentional 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 916 . 2 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → 𝐴𝑅)
2 simpl2 917 . 2 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → 𝐵𝑆)
3 rspc3v.3 . . . 4 (𝑧 = 𝐶 → (𝜃𝜓))
43rspcev 2671 . . 3 ((𝐶𝑇𝜓) → ∃𝑧𝑇 𝜃)
543ad2antl3 1077 . 2 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → ∃𝑧𝑇 𝜃)
6 rspc3v.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜒))
76rexbidv 2342 . . 3 (𝑥 = 𝐴 → (∃𝑧𝑇 𝜑 ↔ ∃𝑧𝑇 𝜒))
8 rspc3v.2 . . . 4 (𝑦 = 𝐵 → (𝜒𝜃))
98rexbidv 2342 . . 3 (𝑦 = 𝐵 → (∃𝑧𝑇 𝜒 ↔ ∃𝑧𝑇 𝜃))
107, 9rspc2ev 2684 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∃𝑧𝑇 𝜃) → ∃𝑥𝑅𝑦𝑆𝑧𝑇 𝜑)
111, 2, 5, 10syl3anc 1144 1 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → ∃𝑥𝑅𝑦𝑆𝑧𝑇 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 894   = wceq 1257  wcel 1407  wrex 2322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-rex 2327  df-v 2574
This theorem is referenced by: (None)
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