MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neleq12d Structured version   Visualization version   GIF version

Theorem neleq12d 3045
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 2821 . . 3 (𝜑 → (𝐴𝐶𝐵𝐷))
43notbid 318 . 2 (𝜑 → (¬ 𝐴𝐶 ↔ ¬ 𝐵𝐷))
5 df-nel 3041 . 2 (𝐴𝐶 ↔ ¬ 𝐴𝐶)
6 df-nel 3041 . 2 (𝐵𝐷 ↔ ¬ 𝐵𝐷)
74, 5, 63bitr4g 314 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1533  wcel 2098  wnel 3040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-cleq 2718  df-clel 2804  df-nel 3041
This theorem is referenced by:  neleq1  3046  neleq2  3047  uhgrspan1  29068  nbgrnself  29124  nbgrnself2  29125  finsumvtxdg2size  29316  fsetsnprcnex  46334
  Copyright terms: Public domain W3C validator