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Theorem rspcdf 42189
 Description: Restricted specialization, using implicit substitution. (Contributed by Emmett Weisz, 16-Jan-2020.)
Hypotheses
Ref Expression
rspcdf.1 𝑥𝜑
rspcdf.2 𝑥𝜒
rspcdf.3 (𝜑𝐴𝐵)
rspcdf.4 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rspcdf (𝜑 → (∀𝑥𝐵 𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rspcdf
StepHypRef Expression
1 rspcdf.1 . . 3 𝑥𝜑
2 rspcdf.4 . . . 4 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
32ex 450 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
41, 3alrimi 2080 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
5 rspcdf.3 . 2 (𝜑𝐴𝐵)
6 rspcdf.2 . . 3 𝑥𝜒
76rspct 3297 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → (∀𝑥𝐵 𝜓𝜒)))
84, 5, 7sylc 65 1 (𝜑 → (∀𝑥𝐵 𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1479   = wceq 1481  Ⅎwnf 1706   ∈ wcel 1988  ∀wral 2909 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-v 3197 This theorem is referenced by: (None)
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