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Theorem rspcimdv 3612
Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcimdv.1 (𝜑𝐴𝐵)
rspcimdv.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rspcimdv (𝜑 → (∀𝑥𝐵 𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcimdv
StepHypRef Expression
1 df-ral 3062 . 2 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
2 rspcimdv.1 . . 3 (𝜑𝐴𝐵)
3 simpr 484 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
43eleq1d 2826 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝑥𝐵𝐴𝐵))
54biimprd 248 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝐴𝐵𝑥𝐵))
6 rspcimdv.2 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
75, 6imim12d 81 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝑥𝐵𝜓) → (𝐴𝐵𝜒)))
82, 7spcimdv 3593 . . 3 (𝜑 → (∀𝑥(𝑥𝐵𝜓) → (𝐴𝐵𝜒)))
92, 8mpid 44 . 2 (𝜑 → (∀𝑥(𝑥𝐵𝜓) → 𝜒))
101, 9biimtrid 242 1 (𝜑 → (∀𝑥𝐵 𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538   = wceq 1540  wcel 2108  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062
This theorem is referenced by:  rspcimedv  3613  rspcdv  3614  wrd2ind  14761  mreexd  17685  mreexexlemd  17687  catcocl  17728  catass  17729  moni  17780  subccocl  17890  funcco  17916  fullfo  17959  fthf1  17964  nati  18003  acsfiindd  18598  chpscmat  22848  mpomulcn  24891  friendshipgt3  30417  lmxrge0  33951  funressnfv  47055  cfsetsnfsetf1  47071
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