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Theorem rspcimedv 3551
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcimdv.1 (𝜑𝐴𝐵)
rspcimedv.2 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
Assertion
Ref Expression
rspcimedv (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcimedv
StepHypRef Expression
1 rspcimdv.1 . . . 4 (𝜑𝐴𝐵)
2 rspcimedv.2 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
32con3d 152 . . . 4 ((𝜑𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒))
41, 3rspcimdv 3550 . . 3 (𝜑 → (∀𝑥𝐵 ¬ 𝜓 → ¬ 𝜒))
54con2d 134 . 2 (𝜑 → (𝜒 → ¬ ∀𝑥𝐵 ¬ 𝜓))
6 dfrex2 3169 . 2 (∃𝑥𝐵 𝜓 ↔ ¬ ∀𝑥𝐵 ¬ 𝜓)
75, 6syl6ibr 251 1 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1539  wcel 2106  wral 3064  wrex 3065
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070
This theorem is referenced by:  rspcedv  3553  scshwfzeqfzo  14537  symgfixfo  19045  slesolex  21829  usgr2pthlem  28128  clwlkclwwlkfo  28370  satfdmlem  33327
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