Description: Deduction from equality to equivalence of membership. (Contributed by NM, 21-Jun-1993) Allow shortening of eleq1 . (Revised by Wolf Lammen, 20-Nov-2019)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | eleq1d.1 | |- ( ph -> A = B ) |
|
| Assertion | eleq1d | |- ( ph -> ( A e. C <-> B e. C ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq1d.1 | |- ( ph -> A = B ) |
|
| 2 | 1 | eqeq2d | |- ( ph -> ( x = A <-> x = B ) ) |
| 3 | 2 | anbi1d | |- ( ph -> ( ( x = A /\ x e. C ) <-> ( x = B /\ x e. C ) ) ) |
| 4 | 3 | exbidv | |- ( ph -> ( E. x ( x = A /\ x e. C ) <-> E. x ( x = B /\ x e. C ) ) ) |
| 5 | dfclel | |- ( A e. C <-> E. x ( x = A /\ x e. C ) ) |
|
| 6 | dfclel | |- ( B e. C <-> E. x ( x = B /\ x e. C ) ) |
|
| 7 | 4 5 6 | 3bitr4g | |- ( ph -> ( A e. C <-> B e. C ) ) |