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Theorem ralprg 3611
Description: Convert a quantification over a pair 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))
Assertion
Ref Expression
ralprg ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ralprg
StepHypRef Expression
1 df-pr 3567 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
21raleqi 2656 . . 3 (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑)
3 ralunb 3288 . . 3 (∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
42, 3bitri 183 . 2 (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
5 ralprg.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
65ralsng 3600 . . 3 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
7 ralprg.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜒))
87ralsng 3600 . . 3 (𝐵𝑊 → (∀𝑥 ∈ {𝐵}𝜑𝜒))
96, 8bi2anan9 596 . 2 ((𝐴𝑉𝐵𝑊) → ((∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (𝜓𝜒)))
104, 9syl5bb 191 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wcel 2128  wral 2435  cun 3100  {csn 3560  {cpr 3561
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-sbc 2938  df-un 3106  df-sn 3566  df-pr 3567
This theorem is referenced by:  raltpg  3613  ralpr  3615  iinxprg  3924  fvinim0ffz  10144  sumpr  11314
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