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Theorem rspcimdv 2657
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 2311 . 2 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
2 rspcimdv.1 . . 3 (𝜑𝐴𝐵)
3 simpr 103 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
43eleq1d 2106 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝑥𝐵𝐴𝐵))
54biimprd 147 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝐴𝐵𝑥𝐵))
6 rspcimdv.2 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
75, 6imim12d 68 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝑥𝐵𝜓) → (𝐴𝐵𝜒)))
82, 7spcimdv 2637 . . 3 (𝜑 → (∀𝑥(𝑥𝐵𝜓) → (𝐴𝐵𝜒)))
92, 8mpid 37 . 2 (𝜑 → (∀𝑥(𝑥𝐵𝜓) → 𝜒))
101, 9syl5bi 141 1 (𝜑 → (∀𝑥𝐵 𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wal 1241   = wceq 1243  wcel 1393  wral 2306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559
This theorem is referenced by:  rspcdv  2659
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