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Theorem neleq12d 2887
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 2682 . . 3 (𝜑 → (𝐴𝐶𝐵𝐷))
43notbid 307 . 2 (𝜑 → (¬ 𝐴𝐶 ↔ ¬ 𝐵𝐷))
5 df-nel 2783 . 2 (𝐴𝐶 ↔ ¬ 𝐴𝐶)
6 df-nel 2783 . 2 (𝐵𝐷 ↔ ¬ 𝐵𝐷)
74, 5, 63bitr4g 302 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195   = wceq 1475  wcel 1977  wnel 2781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606  df-nel 2783
This theorem is referenced by:  neleq1  2888  neleq2  2889  usgrares1  25733  nbgranself  25757  nbgrassovt  25758  frgrancvvdeqlem2  26352  uhgrspan1  40519  nbgrnself  40575  nbgrnself2  40577  frgrncvvdeqlem2  41462
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