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Theorem rspcdf 3594
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 411 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
41, 3alrimi 2202 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
5 rspcdf.3 . 2 (𝜑𝐴𝐵)
6 rspcdf.2 . . 3 𝑥𝜒
76rspct 3593 . 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 205  wa 394  wal 1532   = wceq 1534  wnf 1778  wcel 2099  wral 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-v 3464
This theorem is referenced by:  rspc2daf  32393
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