Description: A version of eqrelrdv . (Contributed by Rodolfo Medina, 10-Oct-2010)
Ref | Expression | ||
---|---|---|---|
Hypothesis | eqrelrdv2.1 | |- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) ) |
|
Assertion | eqrelrdv2 | |- ( ( ( Rel A /\ Rel B ) /\ ph ) -> A = B ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrelrdv2.1 | |- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) ) |
|
2 | 1 | alrimivv | |- ( ph -> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) |
3 | eqrel | |- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) ) |
|
4 | 2 3 | syl5ibr | |- ( ( Rel A /\ Rel B ) -> ( ph -> A = B ) ) |
5 | 4 | imp | |- ( ( ( Rel A /\ Rel B ) /\ ph ) -> A = B ) |