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Theorem ralpr 4700
Description: Convert a restricted universal quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralpr.1 𝐴 ∈ V
ralpr.2 𝐵 ∈ V
ralpr.3 (𝑥 = 𝐴 → (𝜑𝜓))
ralpr.4 (𝑥 = 𝐵 → (𝜑𝜒))
Assertion
Ref Expression
ralpr (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralpr
StepHypRef Expression
1 ralpr.1 . 2 𝐴 ∈ V
2 ralpr.2 . 2 𝐵 ∈ V
3 ralpr.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
4 ralpr.4 . . 3 (𝑥 = 𝐵 → (𝜑𝜒))
53, 4ralprg 4696 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
61, 2, 5mp2an 692 1 (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629
This theorem is referenced by:  fprb  7214  fzprval  13625  fvinim0ffz  13825  wwlktovf1  14996  xpsfrnel  17607  xpsle  17624  isdrs2  18352  pmtrsn  19537  iblcnlem1  25823  lfuhgr1v0e  29271  nbgr2vtx1edg  29367  nbuhgr2vtx1edgb  29369  umgr2v2evd2  29545  2wlklem  29685  dfpth2  29749  2wlkdlem5  29949  2wlkdlem10  29955  clwwlknonex2lem2  30127  3pthdlem1  30183  upgr4cycl4dv4e  30204  subfacp1lem3  35187  poimirlem1  37628  paireqne  47498  requad2  47610  ldepsnlinc  48425  rrx2pnecoorneor  48636  rrx2line  48661  rrx2linest  48663
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