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Theorem spcimdv 2744
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 1830 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
4 spcimdv.1 . 2 (𝜑𝐴𝐵)
5 nfv 1493 . . 3 𝑥𝜒
6 nfcv 2258 . . 3 𝑥𝐴
75, 6spcimgft 2736 . 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 1314   = wceq 1316  wcel 1465
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662
This theorem is referenced by:  spcdv  2745  rspcimdv  2764
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