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Theorem raltp 3781
 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 ↔ (ψ χ θ))
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 3777 . 2 ((A V B V C V) → (x {A, B, C}φ ↔ (ψ χ θ)))
81, 2, 3, 7mp3an 1277 1 (x {A, B, C}φ ↔ (ψ χ θ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2614  Vcvv 2859  {ctp 3739 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-tp 3743 This theorem is referenced by: (None)
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