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 ) ) |
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 ) ) |