Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition ( sb6 ) or a nonfreeness hypothesis ( sb6f ). Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 13-May-1993) Revise df-sb . (Revised by Wolf Lammen, 26-Jul-2023) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sb2 | |- ( A. x ( x = y -> ph ) -> [ y / x ] ph ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm2.27 | |- ( x = y -> ( ( x = y -> ph ) -> ph ) ) |
|
| 2 | 1 | al2imi | |- ( A. x x = y -> ( A. x ( x = y -> ph ) -> A. x ph ) ) |
| 3 | stdpc4 | |- ( A. x ph -> [ y / x ] ph ) |
|
| 4 | 2 3 | syl6 | |- ( A. x x = y -> ( A. x ( x = y -> ph ) -> [ y / x ] ph ) ) |
| 5 | sb4b | |- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) ) |
|
| 6 | 5 | biimprd | |- ( -. A. x x = y -> ( A. x ( x = y -> ph ) -> [ y / x ] ph ) ) |
| 7 | 4 6 | pm2.61i | |- ( A. x ( x = y -> ph ) -> [ y / x ] ph ) |