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Theorem ralpr 4596
 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 4592 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
61, 2, 5mp2an 691 1 (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  {cpr 4527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-sbc 3721  df-un 3886  df-sn 4526  df-pr 4528 This theorem is referenced by:  fprb  6933  fzprval  12963  fvinim0ffz  13151  wwlktovf1  14312  xpsfrnel  16827  xpsle  16844  isdrs2  17541  pmtrsn  18639  iblcnlem1  24391  lfuhgr1v0e  27044  nbgr2vtx1edg  27140  nbuhgr2vtx1edgb  27142  umgr2v2evd2  27317  2wlklem  27457  2wlkdlem5  27715  2wlkdlem10  27721  clwwlknonex2lem2  27893  3pthdlem1  27949  upgr4cycl4dv4e  27970  subfacp1lem3  32542  poimirlem1  35058  paireqne  44026  requad2  44139  ldepsnlinc  44915  rrx2pnecoorneor  45127  rrx2line  45152  rrx2linest  45154
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