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Theorem rspc3ev 2780
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 969 . 2 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → 𝐴𝑅)
2 simpl2 970 . 2 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → 𝐵𝑆)
3 rspc3v.3 . . . 4 (𝑧 = 𝐶 → (𝜃𝜓))
43rspcev 2763 . . 3 ((𝐶𝑇𝜓) → ∃𝑧𝑇 𝜃)
543ad2antl3 1130 . 2 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → ∃𝑧𝑇 𝜃)
6 rspc3v.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜒))
76rexbidv 2415 . . 3 (𝑥 = 𝐴 → (∃𝑧𝑇 𝜑 ↔ ∃𝑧𝑇 𝜒))
8 rspc3v.2 . . . 4 (𝑦 = 𝐵 → (𝜒𝜃))
98rexbidv 2415 . . 3 (𝑦 = 𝐵 → (∃𝑧𝑇 𝜒 ↔ ∃𝑧𝑇 𝜃))
107, 9rspc2ev 2778 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∃𝑧𝑇 𝜃) → ∃𝑥𝑅𝑦𝑆𝑧𝑇 𝜑)
111, 2, 5, 10syl3anc 1201 1 (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → ∃𝑥𝑅𝑦𝑆𝑧𝑇 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 947   = wceq 1316  wcel 1465  wrex 2394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662
This theorem is referenced by: (None)
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