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Theorem rspcdf 3608
 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 415 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
41, 3alrimi 2206 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
5 rspcdf.3 . 2 (𝜑𝐴𝐵)
6 rspcdf.2 . . 3 𝑥𝜒
76rspct 3607 . 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 208   ∧ wa 398  ∀wal 1529   = wceq 1531  Ⅎwnf 1778   ∈ wcel 2108  ∀wral 3136 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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-v 3495 This theorem is referenced by:  rspc2daf  30223
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