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Theorem rspceaimv 3628
Description: Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022.)
Hypothesis
Ref Expression
rspceaimv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rspceaimv ((𝐴𝐵 ∧ ∀𝑦𝐶 (𝜓𝜒)) → ∃𝑥𝐵𝑦𝐶 (𝜑𝜒))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑦)   𝐵(𝑦)   𝐶(𝑦)

Proof of Theorem rspceaimv
StepHypRef Expression
1 rspceaimv.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
21imbi1d 341 . . 3 (𝑥 = 𝐴 → ((𝜑𝜒) ↔ (𝜓𝜒)))
32ralbidv 3176 . 2 (𝑥 = 𝐴 → (∀𝑦𝐶 (𝜑𝜒) ↔ ∀𝑦𝐶 (𝜓𝜒)))
43rspcev 3622 1 ((𝐴𝐵 ∧ ∀𝑦𝐶 (𝜓𝜒)) → ∃𝑥𝐵𝑦𝐶 (𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069
This theorem is referenced by:  brimralrspcev  5209  rexanre  15382  rexico  15389  rlim2lt  15530  rlim3  15531  rlimconst  15577  rlimcn3  15623  reccn2  15630  cn1lem  15631  o1rlimmul  15652  caucvgrlem  15706  divrcnv  15885  chfacffsupp  22878  chfacfscmulfsupp  22881  chfacfpmmulfsupp  22885  tsmsgsum  24163  tsmsres  24168  tsmsxp  24179  metcnpi3  24575  nrginvrcnlem  24728  nghmcn  24782  metdscn  24892  elcncf1di  24935  volcn  25655  itg2cnlem2  25812  abelthlem8  26498  divlogrlim  26692  cxplim  27030  cxploglim  27036  ftalem1  27131  ftalem2  27132  dchrisum0  27579  nmcvcn  30724  blocni  30834  0cnop  32008  0cnfn  32009  idcnop  32010  lnconi  32062  qqhcn  33954  dnicn  36475  ftc1anc  37688  limsupre3uzlem  45691  fourierdlem87  46149
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