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Theorem pm2.61danel 3070
 Description: Deduction eliminating an elementhood in an antecedent. (Contributed by AV, 5-Dec-2021.)
Hypotheses
Ref Expression
pm2.61danel.1 ((𝜑𝐴𝐵) → 𝜓)
pm2.61danel.2 ((𝜑𝐴𝐵) → 𝜓)
Assertion
Ref Expression
pm2.61danel (𝜑𝜓)

Proof of Theorem pm2.61danel
StepHypRef Expression
1 pm2.61danel.1 . 2 ((𝜑𝐴𝐵) → 𝜓)
2 df-nel 3057 . . 3 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
3 pm2.61danel.2 . . 3 ((𝜑𝐴𝐵) → 𝜓)
42, 3sylan2br 598 . 2 ((𝜑 ∧ ¬ 𝐴𝐵) → 𝜓)
51, 4pm2.61dan 813 1 (𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∈ wcel 2112   ∉ wnel 3056 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 210  df-an 401  df-nel 3057 This theorem is referenced by:  clwwlknon1le1  27986  nsnlpligALT  28365  n0lpligALT  28367
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