Description: Specialization, with additional weakening (compared to sp ) to allow bundling of x and y . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017) (Proof shortened by Wolf Lammen, 7-Aug-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | spimfw.1 | |- ( -. ps -> A. x -. ps ) |
|
spimfw.2 | |- ( x = y -> ( ph -> ps ) ) |
||
Assertion | spimfw | |- ( -. A. x -. x = y -> ( A. x ph -> ps ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spimfw.1 | |- ( -. ps -> A. x -. ps ) |
|
2 | spimfw.2 | |- ( x = y -> ( ph -> ps ) ) |
|
3 | 2 | speimfw | |- ( -. A. x -. x = y -> ( A. x ph -> E. x ps ) ) |
4 | df-ex | |- ( E. x ps <-> -. A. x -. ps ) |
|
5 | 1 | con1i | |- ( -. A. x -. ps -> ps ) |
6 | 4 5 | sylbi | |- ( E. x ps -> ps ) |
7 | 3 6 | syl6 | |- ( -. A. x -. x = y -> ( A. x ph -> ps ) ) |