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Theorem rspcedvd 2795
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2793. (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 2793 . 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 1331  wcel 1480  wrex 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688
This theorem is referenced by:  rspcime  2796  rspcedeq1vd  2798  rspcedeq2vd  2799  updjud  6967  modqmuladd  10151  modqmuladdnn0  10153  modfzo0difsn  10180  negfi  11011  divconjdvds  11558  2tp1odd  11592  dfgcd2  11713  qredeu  11789  pw2dvdslemn  11854  xmettx  12693
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