Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-11 . (Revised by Wolf Lammen, 10-May-2023) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | drnfc1.1 | |- ( A. x x = y -> A = B ) |
|
| Assertion | drnfc1 | |- ( A. x x = y -> ( F/_ x A <-> F/_ y B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | drnfc1.1 | |- ( A. x x = y -> A = B ) |
|
| 2 | 1 | eleq2d | |- ( A. x x = y -> ( w e. A <-> w e. B ) ) |
| 3 | 2 | drnf1 | |- ( A. x x = y -> ( F/ x w e. A <-> F/ y w e. B ) ) |
| 4 | 3 | albidv | |- ( A. x x = y -> ( A. w F/ x w e. A <-> A. w F/ y w e. B ) ) |
| 5 | df-nfc | |- ( F/_ x A <-> A. w F/ x w e. A ) |
|
| 6 | df-nfc | |- ( F/_ y B <-> A. w F/ y w e. B ) |
|
| 7 | 4 5 6 | 3bitr4g | |- ( A. x x = y -> ( F/_ x A <-> F/_ y B ) ) |