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Theorem raltpg 3624
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 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
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 3622 . . . 4 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
4 raltpg.3 . . . . 5 (𝑥 = 𝐶 → (𝜑𝜃))
54ralsng 3611 . . . 4 (𝐶𝑋 → (∀𝑥 ∈ {𝐶}𝜑𝜃))
63, 5bi2anan9 596 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → ((∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∧ 𝜃)))
763impa 1183 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∧ 𝜃)))
8 df-tp 3579 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
98raleqi 2663 . . 3 (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ ∀𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑)
10 ralunb 3299 . . 3 (∀𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑 ↔ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑))
119, 10bitri 183 . 2 (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑))
12 df-3an 969 . 2 ((𝜓𝜒𝜃) ↔ ((𝜓𝜒) ∧ 𝜃))
137, 11, 123bitr4g 222 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 967   = wceq 1342  wcel 2135  wral 2442  cun 3110  {csn 3571  {cpr 3572  {ctp 3573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-v 2724  df-sbc 2948  df-un 3116  df-sn 3577  df-pr 3578  df-tp 3579
This theorem is referenced by:  raltp  3628  sumtp  11345
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