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Theorem rspcedvd 2729
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2727. (Contributed by AV, 27-Nov-2019.)
Hypotheses
Ref Expression
rspcedvd.1 (𝜑𝐴𝐵)
rspcedvd.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
rspcedvd.3 (𝜑𝜒)
Assertion
Ref Expression
rspcedvd (𝜑 → ∃𝑥𝐵 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcedvd
StepHypRef Expression
1 rspcedvd.3 . 2 (𝜑𝜒)
2 rspcedvd.1 . . 3 (𝜑𝐴𝐵)
3 rspcedvd.2 . . 3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
42, 3rspcedv 2727 . 2 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
51, 4mpd 13 1 (𝜑 → ∃𝑥𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1290  wcel 1439  wrex 2361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622
This theorem is referenced by:  rspcedeq1vd  2731  rspcedeq2vd  2732  updjud  6827  modqmuladd  9834  modqmuladdnn0  9836  modfzo0difsn  9863  negfi  10720  divconjdvds  11189  2tp1odd  11223  dfgcd2  11342  qredeu  11418  pw2dvdslemn  11482
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