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Theorem raltp 3454
Description: Convert a quantification over a 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 3450 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
81, 2, 3, 7mp3an 1243 1 (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  w3a 896   = wceq 1259  wcel 1409  wral 2323  Vcvv 2574  {ctp 3404
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 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-sbc 2787  df-un 2949  df-sn 3408  df-pr 3409  df-tp 3410
This theorem is referenced by:  fztpval  9046
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