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Theorem nelcon3d 2895
Description: Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020.)
Hypothesis
Ref Expression
nelcon3d.1 (𝜑 → (𝐴𝐵𝐶𝐷))
Assertion
Ref Expression
nelcon3d (𝜑 → (𝐶𝐷𝐴𝐵))

Proof of Theorem nelcon3d
StepHypRef Expression
1 nelcon3d.1 . . 3 (𝜑 → (𝐴𝐵𝐶𝐷))
21con3d 147 . 2 (𝜑 → (¬ 𝐶𝐷 → ¬ 𝐴𝐵))
3 df-nel 2783 . 2 (𝐶𝐷 ↔ ¬ 𝐶𝐷)
4 df-nel 2783 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
52, 3, 43imtr4g 284 1 (𝜑 → (𝐶𝐷𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1977  wnel 2781
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 196  df-nel 2783
This theorem is referenced by:  lcmfnnval  15124  isnmgm  17018  prcssprc  40101  mgmplusfreseq  41555
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