Description: A theorem used in elimination of disjoint variable restriction on x and y by replacing it with a distinctor -. A. x x = z . (Contributed by NM, 15-May-1993) Proof is based on wl-sbalnae now. See also sbal1 . (Revised by Wolf Lammen, 25-Jul-2019) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | wl-sbal1 | |- ( -. A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | naev | |- ( -. A. x x = z -> -. A. x x = y ) |
|
| 2 | wl-sbalnae | |- ( ( -. A. x x = y /\ -. A. x x = z ) -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) |
|
| 3 | 1 2 | mpancom | |- ( -. A. x x = z -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) |