Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition ( sb5 ) or a nonfreeness hypothesis ( sb5f ). Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sb1v when possible. (Contributed by NM, 13-May-1993) Revise df-sb . (Revised by Wolf Lammen, 21-Feb-2024) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sb1 | |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | spsbe | |- ( [ y / x ] ph -> E. x ph ) |
|
| 2 | pm3.2 | |- ( x = y -> ( ph -> ( x = y /\ ph ) ) ) |
|
| 3 | 2 | aleximi | |- ( A. x x = y -> ( E. x ph -> E. x ( x = y /\ ph ) ) ) |
| 4 | 1 3 | syl5 | |- ( A. x x = y -> ( [ y / x ] ph -> E. x ( x = y /\ ph ) ) ) |
| 5 | sb3b | |- ( -. A. x x = y -> ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) ) |
|
| 6 | 5 | biimpd | |- ( -. A. x x = y -> ( [ y / x ] ph -> E. x ( x = y /\ ph ) ) ) |
| 7 | 4 6 | pm2.61i | |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) ) |