MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spcimedv Structured version   Visualization version   GIF version

Theorem spcimedv 3594
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 152 . . . 4 ((𝜑𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒))
41, 3spcimdv 3592 . . 3 (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒))
54con2d 134 . 2 (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓))
6 df-ex 1779 . 2 (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
75, 6imbitrrdi 252 1 (𝜑 → (𝜒 → ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1537   = wceq 1539  wex 1778  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-clel 2815
This theorem is referenced by:  spc3egv  3602  hashf1rn  14392  cshwsexaOLD  14864  wwlktovfo  14998  uvcendim  21868  wlkiswwlks2  29896  wwlksnextsurj  29921  elwwlks2  29987  elwspths2spth  29988  clwlkclwwlklem1  30019  sticksstones4  42151  rtrclex  43635  clcnvlem  43641  iunrelexpuztr  43737
  Copyright terms: Public domain W3C validator