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Theorem spcimedv 3598
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 155 . . . 4 ((𝜑𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒))
41, 3spcimdv 3596 . . 3 (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒))
54con2d 136 . 2 (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓))
6 df-ex 1774 . 2 (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
75, 6syl6ibr 253 1 (𝜑 → (𝜒 → ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1528   = wceq 1530  wex 1773  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-cleq 2817  df-clel 2897
This theorem is referenced by:  spc3egv  3607  hashf1rn  13706  cshwsexa  14179  wwlktovfo  14315  uvcendim  20907  wlkiswwlks2  27568  wwlksnextsurj  27593  elwwlks2  27660  elwspths2spth  27661  clwlkclwwlklem1  27692  rtrclex  39839  clcnvlem  39845  iunrelexpuztr  39926
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