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Theorem spcimdv 3511
 Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) Avoid ax-10 2143 and ax-11 2159. (Revised by Gino Giotto, 20-Aug-2023.)
Hypotheses
Ref Expression
spcimdv.1 (𝜑𝐴𝐵)
spcimdv.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
spcimdv (𝜑 → (∀𝑥𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem spcimdv
StepHypRef Expression
1 spcimdv.1 . . . 4 (𝜑𝐴𝐵)
2 elisset 2834 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
31, 2syl 17 . . 3 (𝜑 → ∃𝑥 𝑥 = 𝐴)
4 spcimdv.2 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
54ex 417 . . . 4 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
65eximdv 1919 . . 3 (𝜑 → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓𝜒)))
73, 6mpd 15 . 2 (𝜑 → ∃𝑥(𝜓𝜒))
8 19.36v 1995 . 2 (∃𝑥(𝜓𝜒) ↔ (∀𝑥𝜓𝜒))
97, 8sylib 221 1 (𝜑 → (∀𝑥𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-clel 2831 This theorem is referenced by:  spcdv  3512  spcimedv  3513  rspcimdv  3532  mrieqv2d  16969  mreexexlemd  16974  intabssd  40601
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