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Theorem rspc3dv 3624
Description: 3-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2025.)
Hypotheses
Ref Expression
rspc3dv.1 (𝑥 = 𝐴 → (𝜓𝜃))
rspc3dv.2 (𝑦 = 𝐵 → (𝜃𝜏))
rspc3dv.3 (𝑧 = 𝐶 → (𝜏𝜒))
rspc3dv.4 (𝜑 → ∀𝑥𝐷𝑦𝐸𝑧𝐹 𝜓)
rspc3dv.5 (𝜑𝐴𝐷)
rspc3dv.6 (𝜑𝐵𝐸)
rspc3dv.7 (𝜑𝐶𝐹)
Assertion
Ref Expression
rspc3dv (𝜑𝜒)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝐷   𝑥,𝐸,𝑦   𝑥,𝐹,𝑦,𝑧   𝜒,𝑧   𝜏,𝑦   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦)   𝜃(𝑦,𝑧)   𝜏(𝑥,𝑧)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑦,𝑧)   𝐸(𝑧)

Proof of Theorem rspc3dv
StepHypRef Expression
1 rspc3dv.5 . . 3 (𝜑𝐴𝐷)
2 rspc3dv.6 . . 3 (𝜑𝐵𝐸)
3 rspc3dv.7 . . 3 (𝜑𝐶𝐹)
41, 2, 33jca 1125 . 2 (𝜑 → (𝐴𝐷𝐵𝐸𝐶𝐹))
5 rspc3dv.4 . 2 (𝜑 → ∀𝑥𝐷𝑦𝐸𝑧𝐹 𝜓)
6 rspc3dv.1 . . 3 (𝑥 = 𝐴 → (𝜓𝜃))
7 rspc3dv.2 . . 3 (𝑦 = 𝐵 → (𝜃𝜏))
8 rspc3dv.3 . . 3 (𝑧 = 𝐶 → (𝜏𝜒))
96, 7, 8rspc3v 3622 . 2 ((𝐴𝐷𝐵𝐸𝐶𝐹) → (∀𝑥𝐷𝑦𝐸𝑧𝐹 𝜓𝜒))
104, 5, 9sylc 65 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  wral 3055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056
This theorem is referenced by:  mulsasslem3  28016  domnlcan  32880
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