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Theorem neleq12d 3122
Description: Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
Hypotheses
Ref Expression
neleq12d.1 (𝜑𝐴 = 𝐵)
neleq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
neleq12d (𝜑 → (𝐴𝐶𝐵𝐷))

Proof of Theorem neleq12d
StepHypRef Expression
1 neleq12d.1 . . . 4 (𝜑𝐴 = 𝐵)
2 neleq12d.2 . . . 4 (𝜑𝐶 = 𝐷)
31, 2eleq12d 2910 . . 3 (𝜑 → (𝐴𝐶𝐵𝐷))
43notbid 321 . 2 (𝜑 → (¬ 𝐴𝐶 ↔ ¬ 𝐵𝐷))
5 df-nel 3119 . 2 (𝐴𝐶 ↔ ¬ 𝐴𝐶)
6 df-nel 3119 . 2 (𝐵𝐷 ↔ ¬ 𝐵𝐷)
74, 5, 63bitr4g 317 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1538  wcel 2115  wnel 3118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817  df-clel 2896  df-nel 3119
This theorem is referenced by:  neleq1  3123  neleq2  3124  uhgrspan1  27096  nbgrnself  27152  nbgrnself2  27153  finsumvtxdg2size  27343
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