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Theorem spcimdv 2693
Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimdv.1 (𝜑𝐴𝐵)
spcimdv.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
spcimdv (𝜑 → (∀𝑥𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem spcimdv
StepHypRef Expression
1 spcimdv.2 . . . 4 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
21ex 113 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
32alrimiv 1797 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
4 spcimdv.1 . 2 (𝜑𝐴𝐵)
5 nfv 1462 . . 3 𝑥𝜒
6 nfcv 2223 . . 3 𝑥𝐴
75, 6spcimgft 2685 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → (∀𝑥𝜓𝜒)))
83, 4, 7sylc 61 1 (𝜑 → (∀𝑥𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283   = wceq 1285  wcel 1434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614
This theorem is referenced by:  spcdv  2694  rspcimdv  2713
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