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Theorem neleq12d 2384
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 2382 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
31, 2syl 14 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
4 neleq12d.2 . . 3 (𝜑𝐶 = 𝐷)
5 neleq2 2383 . . 3 (𝐶 = 𝐷 → (𝐵𝐶𝐵𝐷))
64, 5syl 14 . 2 (𝜑 → (𝐵𝐶𝐵𝐷))
73, 6bitrd 187 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wnel 2378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-cleq 2108  df-clel 2111  df-nel 2379
This theorem is referenced by: (None)
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