Metamath Proof Explorer


Theorem brfvid

Description: If two elements are connected by a value of the identity relation, then they are connected via the argument. (Contributed by RP, 21-Jul-2020)

Ref Expression
Hypothesis brfvid.r
|- ( ph -> R e. _V )
Assertion brfvid
|- ( ph -> ( A ( _I ` R ) B <-> A R B ) )

Proof

Step Hyp Ref Expression
1 brfvid.r
 |-  ( ph -> R e. _V )
2 fvi
 |-  ( R e. _V -> ( _I ` R ) = R )
3 1 2 syl
 |-  ( ph -> ( _I ` R ) = R )
4 3 breqd
 |-  ( ph -> ( A ( _I ` R ) B <-> A R B ) )