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Theorem pm2.61danel 2940
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 2927 . . 3 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
3 pm2.61danel.2 . . 3 ((𝜑𝐴𝐵) → 𝜓)
42, 3sylan2br 492 . 2 ((𝜑 ∧ ¬ 𝐴𝐵) → 𝜓)
51, 4pm2.61dan 849 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wcel 2030  wnel 2926
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 197  df-an 385  df-nel 2927
This theorem is referenced by:  clwwlknon1le1  27076  nsnlpligALT  27464  n0lpligALT  27466
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