Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind is derived from e2ebindVD . (Contributed by Alan Sare, 27-Nov-2014) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | e2ebind | |- ( A. x x = y -> ( E. x E. y ph <-> E. y ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | biidd | |- ( A. y y = x -> ( ph <-> ph ) ) |
|
| 2 | 1 | drex1 | |- ( A. y y = x -> ( E. y ph <-> E. x ph ) ) |
| 3 | 2 | drex2 | |- ( A. y y = x -> ( E. y E. y ph <-> E. y E. x ph ) ) |
| 4 | excom | |- ( E. y E. x ph <-> E. x E. y ph ) |
|
| 5 | 3 4 | bitrdi | |- ( A. y y = x -> ( E. y E. y ph <-> E. x E. y ph ) ) |
| 6 | nfe1 | |- F/ y E. y ph |
|
| 7 | 6 | 19.9 | |- ( E. y E. y ph <-> E. y ph ) |
| 8 | 5 7 | bitr3di | |- ( A. y y = x -> ( E. x E. y ph <-> E. y ph ) ) |
| 9 | 8 | aecoms | |- ( A. x x = y -> ( E. x E. y ph <-> E. y ph ) ) |