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Theorem ralprg 4424
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 4371 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
21raleqi 3325 . . 3 (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑)
3 ralunb 3992 . . 3 (∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
42, 3bitri 267 . 2 (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
5 ralprg.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
65ralsng 4409 . . 3 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
7 ralprg.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜒))
87ralsng 4409 . . 3 (𝐵𝑊 → (∀𝑥 ∈ {𝐵}𝜑𝜒))
96, 8bi2anan9 630 . 2 ((𝐴𝑉𝐵𝑊) → ((∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (𝜓𝜒)))
104, 9syl5bb 275 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  cun 3767  {csn 4368  {cpr 4370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387  df-sbc 3634  df-un 3774  df-sn 4369  df-pr 4371
This theorem is referenced by:  raltpg  4426  ralpr  4428  iinxprg  4791  disjprg  4839  fpropnf1  6752  f12dfv  6757  f13dfv  6758  suppr  8619  infpr  8651  pfx2  14032  sumpr  14818  gcdcllem2  15557  lcmfpr  15675  joinval2lem  17323  meetval2lem  17337  sgrp2rid2  17729  sgrp2nmndlem4  17731  sgrp2nmndlem5  17732  iccntr  22952  limcun  24000  cplgr3v  26685  3wlkdlem4  27506  frgr3v  27624  3vfriswmgr  27627  prsiga  30710
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