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Theorem neleq12d 2352
Description: Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)
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 . . 3 (𝜑𝐴 = 𝐵)
2 neleq1 2350 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
31, 2syl 14 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
4 neleq12d.2 . . 3 (𝜑𝐶 = 𝐷)
5 neleq2 2351 . . 3 (𝐶 = 𝐷 → (𝐵𝐶𝐵𝐷))
64, 5syl 14 . 2 (𝜑 → (𝐵𝐶𝐵𝐷))
73, 6bitrd 186 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1287  wnel 2346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-cleq 2078  df-clel 2081  df-nel 2347
This theorem is referenced by: (None)
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