Metamath Proof Explorer


Theorem neleq12d

Description: Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016) (Proof shortened by Wolf Lammen, 25-Nov-2019)

Ref Expression
Hypotheses neleq12d.1
|- ( ph -> A = B )
neleq12d.2
|- ( ph -> C = D )
Assertion neleq12d
|- ( ph -> ( A e/ C <-> B e/ D ) )

Proof

Step Hyp Ref Expression
1 neleq12d.1
 |-  ( ph -> A = B )
2 neleq12d.2
 |-  ( ph -> C = D )
3 1 2 eleq12d
 |-  ( ph -> ( A e. C <-> B e. D ) )
4 3 notbid
 |-  ( ph -> ( -. A e. C <-> -. B e. D ) )
5 df-nel
 |-  ( A e/ C <-> -. A e. C )
6 df-nel
 |-  ( B e/ D <-> -. B e. D )
7 4 5 6 3bitr4g
 |-  ( ph -> ( A e/ C <-> B e/ D ) )