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Theorem spcimedv 3323
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 148 . . . 4 ((𝜑𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒))
41, 3spcimdv 3321 . . 3 (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒))
54con2d 129 . 2 (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓))
6 df-ex 1745 . 2 (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
75, 6syl6ibr 242 1 (𝜑 → (𝜒 → ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233
This theorem is referenced by:  hashf1rn  13181  cshwsexa  13616  wwlktovfo  13747  uvcendim  20234  wlkiswwlks2  26829  wlknwwlksnsur  26844  wlkwwlksur  26851  wwlksnextsur  26863  elwwlks2  26933  elwspths2spth  26934  clwlkclwwlklem1  26965  rtrclex  38241  clcnvlem  38247  iunrelexpuztr  38328
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