Description: Transfer existence from a variable x to another variable y contained in expression A . (Contributed by NM, 10-Jun-2005) (Revised by Mario Carneiro, 15-Aug-2014)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ralxfr.1 | |- ( y e. C -> A e. B ) |
|
ralxfr.2 | |- ( x e. B -> E. y e. C x = A ) |
||
ralxfr.3 | |- ( x = A -> ( ph <-> ps ) ) |
||
Assertion | rexxfr | |- ( E. x e. B ph <-> E. y e. C ps ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralxfr.1 | |- ( y e. C -> A e. B ) |
|
2 | ralxfr.2 | |- ( x e. B -> E. y e. C x = A ) |
|
3 | ralxfr.3 | |- ( x = A -> ( ph <-> ps ) ) |
|
4 | dfrex2 | |- ( E. x e. B ph <-> -. A. x e. B -. ph ) |
|
5 | dfrex2 | |- ( E. y e. C ps <-> -. A. y e. C -. ps ) |
|
6 | 3 | notbid | |- ( x = A -> ( -. ph <-> -. ps ) ) |
7 | 1 2 6 | ralxfr | |- ( A. x e. B -. ph <-> A. y e. C -. ps ) |
8 | 5 7 | xchbinxr | |- ( E. y e. C ps <-> -. A. x e. B -. ph ) |
9 | 4 8 | bitr4i | |- ( E. x e. B ph <-> E. y e. C ps ) |