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Theorem raltpg 3777
 Description: Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralprg.1 (x = A → (φψ))
ralprg.2 (x = B → (φχ))
raltpg.3 (x = C → (φθ))
Assertion
Ref Expression
raltpg ((A V B W C X) → (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 hints:   φ(x)   V(x)   W(x)   X(x)

Proof of Theorem raltpg
StepHypRef Expression
1 ralprg.1 . . . . 5 (x = A → (φψ))
2 ralprg.2 . . . . 5 (x = B → (φχ))
31, 2ralprg 3775 . . . 4 ((A V B W) → (x {A, B}φ ↔ (ψ χ)))
4 raltpg.3 . . . . 5 (x = C → (φθ))
54ralsng 3765 . . . 4 (C X → (x {C}φθ))
63, 5bi2anan9 843 . . 3 (((A V B W) C X) → ((x {A, B}φ x {C}φ) ↔ ((ψ χ) θ)))
763impa 1146 . 2 ((A V B W C X) → ((x {A, B}φ x {C}φ) ↔ ((ψ χ) θ)))
8 df-tp 3743 . . . 4 {A, B, C} = ({A, B} ∪ {C})
98raleqi 2811 . . 3 (x {A, B, C}φx ({A, B} ∪ {C})φ)
10 ralunb 3444 . . 3 (x ({A, B} ∪ {C})φ ↔ (x {A, B}φ x {C}φ))
119, 10bitri 240 . 2 (x {A, B, C}φ ↔ (x {A, B}φ x {C}φ))
12 df-3an 936 . 2 ((ψ χ θ) ↔ ((ψ χ) θ))
137, 11, 123bitr4g 279 1 ((A V B W C X) → (x {A, B, C}φ ↔ (ψ χ θ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ∪ cun 3207  {csn 3737  {cpr 3738  {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:  raltp  3781
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