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Theorem rspcda 2678
 Description: Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 29-Jun-2020.)
Hypotheses
Ref Expression
rspcdva.1 (𝑥 = 𝐶 → (𝜓𝜒))
rspcdva.2 (𝜑 → ∀𝑥𝐴 𝜓)
rspcdva.3 (𝜑𝐶𝐴)
rspcda.1 𝑥𝜑
Assertion
Ref Expression
rspcda (𝜑𝜒)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rspcda
StepHypRef Expression
1 rspcdva.3 . 2 (𝜑𝐶𝐴)
2 rspcdva.2 . 2 (𝜑 → ∀𝑥𝐴 𝜓)
3 rspcdva.1 . . 3 (𝑥 = 𝐶 → (𝜓𝜒))
43rspcv 2669 . 2 (𝐶𝐴 → (∀𝑥𝐴 𝜓𝜒))
51, 2, 4sylc 60 1 (𝜑𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   = wceq 1259  Ⅎwnf 1365   ∈ wcel 1409  ∀wral 2323 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576 This theorem is referenced by: (None)
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