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Theorem rspce 3611
Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
Hypotheses
Ref Expression
rspc.1 𝑥𝜓
rspc.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rspce ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rspce
StepHypRef Expression
1 nfcv 2903 . . . 4 𝑥𝐴
2 nfv 1912 . . . . 5 𝑥 𝐴𝐵
3 rspc.1 . . . . 5 𝑥𝜓
42, 3nfan 1897 . . . 4 𝑥(𝐴𝐵𝜓)
5 eleq1 2827 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
6 rspc.2 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6anbi12d 632 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
81, 4, 7spcegf 3592 . . 3 (𝐴𝐵 → ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑)))
98anabsi5 669 . 2 ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑))
10 df-rex 3069 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
119, 10sylibr 234 1 ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wnf 1780  wcel 2106  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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rex 3069
This theorem is referenced by:  reuop  6315  ac6c4  10519  infcvgaux1i  15890  iunmbl2  25606  gsumpart  33043  esumcvg  34067  ptrecube  37607  poimirlem24  37631  sdclem1  37730  uzwo4  44993  eliuniincex  45049  elrnmpt1sf  45132  iuneqfzuzlem  45284  uzublem  45380  uzub  45381  limsupubuzlem  45668  sge0gerp  46351  smflim  46733  reupr  47447
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