Description: Deduction form of rabeq . Note that contrary to rabeq it has no disjoint variable condition. (Contributed by BJ, 27-Apr-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | bj-rabeqd.nf | |- F/ x ph |
|
bj-rabeqd.1 | |- ( ph -> A = B ) |
||
Assertion | bj-rabeqd | |- ( ph -> { x e. A | ps } = { x e. B | ps } ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-rabeqd.nf | |- F/ x ph |
|
2 | bj-rabeqd.1 | |- ( ph -> A = B ) |
|
3 | eleq2 | |- ( A = B -> ( x e. A <-> x e. B ) ) |
|
4 | 3 | anbi1d | |- ( A = B -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ps ) ) ) |
5 | 2 4 | syl | |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ps ) ) ) |
6 | 1 5 | bj-rabbida2 | |- ( ph -> { x e. A | ps } = { x e. B | ps } ) |