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Theorem spcimedv 3444
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 3442 . . 3 (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒))
54con2d 131 . 2 (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓))
6 df-ex 1875 . 2 (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
75, 6syl6ibr 243 1 (𝜑 → (𝜒 → ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1650   = wceq 1652  wex 1874  wcel 2155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352
This theorem is referenced by:  hashf1rn  13345  cshwsexa  13855  wwlktovfo  13990  uvcendim  20462  wlkiswwlks2  27065  wlknwwlksnsurOLD  27080  wlkwwlksurOLD  27088  wwlksnextsur  27100  elwwlks2  27171  elwspths2spth  27172  clwlkclwwlklem1  27205  rtrclex  38531  clcnvlem  38537  iunrelexpuztr  38618
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