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Theorem spcimedv 3443
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimdv.1 (𝜑𝐴𝐵)
spcimedv.2 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
Assertion
Ref Expression
spcimedv (𝜑 → (𝜒 → ∃𝑥𝜓))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem spcimedv
StepHypRef Expression
1 spcimdv.1 . . . 4 (𝜑𝐴𝐵)
2 spcimedv.2 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
32con3d 149 . . . 4 ((𝜑𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒))
41, 3spcimdv 3441 . . 3 (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒))
54con2d 131 . 2 (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓))
6 df-ex 1853 . 2 (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
75, 6syl6ibr 242 1 (𝜑 → (𝜒 → ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wal 1629   = wceq 1631  wex 1852  wcel 2145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353
This theorem is referenced by:  hashf1rn  13345  cshwsexa  13779  wwlktovfo  13911  uvcendim  20403  wlkiswwlks2  27009  wlknwwlksnsurOLD  27024  wlkwwlksurOLD  27032  wwlksnextsur  27044  elwwlks2  27115  elwspths2spth  27116  clwlkclwwlklem1  27149  rtrclex  38450  clcnvlem  38456  iunrelexpuztr  38537
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