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Theorem rspcimedv 3513
 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 150 . . . 4 ((𝜑𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒))
41, 3rspcimdv 3512 . . 3 (𝜑 → (∀𝑥𝐵 ¬ 𝜓 → ¬ 𝜒))
54con2d 132 . 2 (𝜑 → (𝜒 → ¬ ∀𝑥𝐵 ¬ 𝜓))
6 dfrex2 3177 . 2 (∃𝑥𝐵 𝜓 ↔ ¬ ∀𝑥𝐵 ¬ 𝜓)
75, 6syl6ibr 244 1 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400 This theorem is referenced by:  rspcedv  3515  scshwfzeqfzo  13981  symgfixfo  18246  slesolex  20898  usgr2pthlem  27119  clwlkclwwlkfoOLD  27397  clwlkclwwlkfo  27401
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