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Theorem rspcimdv 3530
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 3087 . 2 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
2 rspcimdv.1 . . 3 (𝜑𝐴𝐵)
3 simpr 477 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
43eleq1d 2844 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝑥𝐵𝐴𝐵))
54biimprd 240 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝐴𝐵𝑥𝐵))
6 rspcimdv.2 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
75, 6imim12d 81 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝑥𝐵𝜓) → (𝐴𝐵𝜒)))
82, 7spcimdv 3505 . . 3 (𝜑 → (∀𝑥(𝑥𝐵𝜓) → (𝐴𝐵𝜒)))
92, 8mpid 44 . 2 (𝜑 → (∀𝑥(𝑥𝐵𝜓) → 𝜒))
101, 9syl5bi 234 1 (𝜑 → (∀𝑥𝐵 𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wal 1505   = wceq 1507  wcel 2050  wral 3082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-ral 3087  df-v 3411
This theorem is referenced by:  rspcimedv  3531  rspcdv  3532  wrd2ind  13911  wrd2indOLD  13912  mreexd  16765  mreexexlemd  16767  catcocl  16808  catass  16809  moni  16858  subccocl  16967  funcco  16993  fullfo  17034  fthf1  17039  nati  17077  acsfiindd  17639  chpscmat  21148  friendshipgt3  27949  lmxrge0  30839  funressnfv  42683
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