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Theorem raltp 4702
Description: Convert a universal quantification over an unordered triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
raltp.1 𝐴 ∈ V
raltp.2 𝐵 ∈ V
raltp.3 𝐶 ∈ V
raltp.4 (𝑥 = 𝐴 → (𝜑𝜓))
raltp.5 (𝑥 = 𝐵 → (𝜑𝜒))
raltp.6 (𝑥 = 𝐶 → (𝜑𝜃))
Assertion
Ref Expression
raltp (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem raltp
StepHypRef Expression
1 raltp.1 . 2 𝐴 ∈ V
2 raltp.2 . 2 𝐵 ∈ V
3 raltp.3 . 2 𝐶 ∈ V
4 raltp.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
5 raltp.5 . . 3 (𝑥 = 𝐵 → (𝜑𝜒))
6 raltp.6 . . 3 (𝑥 = 𝐶 → (𝜑𝜃))
74, 5, 6raltpg 4695 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
81, 2, 3, 7mp3an 1457 1 (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  wral 3053  Vcvv 3466  {ctp 4625
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 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-v 3468  df-un 3946  df-sn 4622  df-pr 4624  df-tp 4626
This theorem is referenced by:  fztpval  13564  2wlkdlem4  29676  2pthdlem1  29678  3wlkdlem5  29910  3wlkdlem10  29916  upgr3v3e3cycl  29927  poimirlem9  37000
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