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Theorem spcimdv 2774
 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 114 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
32alrimiv 1847 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
4 spcimdv.1 . 2 (𝜑𝐴𝐵)
5 nfv 1509 . . 3 𝑥𝜒
6 nfcv 2282 . . 3 𝑥𝐴
75, 6spcimgft 2766 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → (∀𝑥𝜓𝜒)))
83, 4, 7sylc 62 1 (𝜑 → (∀𝑥𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330   = wceq 1332   ∈ wcel 1481 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692 This theorem is referenced by:  spcdv  2775  rspcimdv  2795
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