Description: Move quantifier in and out of substitution. Revised to remove a distinct variable constraint. (Contributed by NM, 2-Jan-2002) Proof is based on wl-sbalnae now. See also sbal2 . (Revised by Wolf Lammen, 25-Jul-2019) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | wl-sbal2 | |- ( -. A. x x = y -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | naev | |- ( -. A. x x = y -> -. A. x x = z ) |
|
| 2 | wl-sbalnae | |- ( ( -. A. x x = y /\ -. A. x x = z ) -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) |
|
| 3 | 1 2 | mpdan | |- ( -. A. x x = y -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) |