Description: Alternate elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020)
Ref | Expression | ||
---|---|---|---|
Hypothesis | ceqsralv2.1 | |- ( x = A -> ( ph <-> ps ) ) |
|
Assertion | ceqsralv2 | |- ( A. x e. B ( x = A -> ph ) <-> ( A e. B -> ps ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ceqsralv2.1 | |- ( x = A -> ( ph <-> ps ) ) |
|
2 | 1 | notbid | |- ( x = A -> ( -. ph <-> -. ps ) ) |
3 | 2 | ceqsrexv2 | |- ( E. x e. B ( x = A /\ -. ph ) <-> ( A e. B /\ -. ps ) ) |
4 | rexanali | |- ( E. x e. B ( x = A /\ -. ph ) <-> -. A. x e. B ( x = A -> ph ) ) |
|
5 | annim | |- ( ( A e. B /\ -. ps ) <-> -. ( A e. B -> ps ) ) |
|
6 | 3 4 5 | 3bitr3i | |- ( -. A. x e. B ( x = A -> ph ) <-> -. ( A e. B -> ps ) ) |
7 | 6 | con4bii | |- ( A. x e. B ( x = A -> ph ) <-> ( A e. B -> ps ) ) |