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Theorem rspcimdv 2745
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 2380 . 2 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
2 rspcimdv.1 . . 3 (𝜑𝐴𝐵)
3 simpr 109 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
43eleq1d 2168 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝑥𝐵𝐴𝐵))
54biimprd 157 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝐴𝐵𝑥𝐵))
6 rspcimdv.2 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
75, 6imim12d 74 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝑥𝐵𝜓) → (𝐴𝐵𝜒)))
82, 7spcimdv 2725 . . 3 (𝜑 → (∀𝑥(𝑥𝐵𝜓) → (𝐴𝐵𝜒)))
92, 8mpid 42 . 2 (𝜑 → (∀𝑥(𝑥𝐵𝜓) → 𝜒))
101, 9syl5bi 151 1 (𝜑 → (∀𝑥𝐵 𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1297   = wceq 1299  wcel 1448  wral 2375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643
This theorem is referenced by:  rspcdv  2747
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