Description: We can always find values matching x and y , as long as they are represented by distinct variables. Version of 2ax6elem with a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Wolf Lammen, 28-Sep-2018) (Proof shortened by Wolf Lammen, 3-Oct-2023) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | 2ax6e | |- E. z E. w ( z = x /\ w = y ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | aeveq | |- ( A. w w = z -> z = x ) |
|
| 2 | aeveq | |- ( A. w w = z -> w = y ) |
|
| 3 | 1 2 | jca | |- ( A. w w = z -> ( z = x /\ w = y ) ) |
| 4 | 3 | 19.8ad | |- ( A. w w = z -> E. w ( z = x /\ w = y ) ) |
| 5 | 4 | 19.8ad | |- ( A. w w = z -> E. z E. w ( z = x /\ w = y ) ) |
| 6 | 2ax6elem | |- ( -. A. w w = z -> E. z E. w ( z = x /\ w = y ) ) |
|
| 7 | 5 6 | pm2.61i | |- E. z E. w ( z = x /\ w = y ) |