Description: Alternate proof of eleq2w2 and special instance of eleq2 . (Contributed by BJ, 22-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | eleq2w2ALT | |- ( A = B -> ( x e. A <-> x e. B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfcleq | |- ( A = B <-> A. y ( y e. A <-> y e. B ) ) |
|
| 2 | 1 | biimpi | |- ( A = B -> A. y ( y e. A <-> y e. B ) ) |
| 3 | eleq1w | |- ( y = x -> ( y e. A <-> x e. A ) ) |
|
| 4 | eleq1w | |- ( y = x -> ( y e. B <-> x e. B ) ) |
|
| 5 | 3 4 | bibi12d | |- ( y = x -> ( ( y e. A <-> y e. B ) <-> ( x e. A <-> x e. B ) ) ) |
| 6 | 5 | spvv | |- ( A. y ( y e. A <-> y e. B ) -> ( x e. A <-> x e. B ) ) |
| 7 | 2 6 | syl | |- ( A = B -> ( x e. A <-> x e. B ) ) |