Let and be two forcing notions such that given any generic subset of there is a generic subset of with and vice-versa. Then and are equivalent.
Since if , for any -name , it follows that if and then .
![$ \mathfrak{M}[G]=\mathfrak{M}[H]$](http://images.planetmath.org:8080/cache/objects/3257/l2h/img7.png "$ \mathfrak{M}[G]=\mathfrak{M}[H]$"){kind=small}
![$ G\in\mathfrak{M}[H]$](http://images.planetmath.org:8080/cache/objects/3257/l2h/img10.png "$ G\in\mathfrak{M}[H]$"){kind=small}
![$ \tau[G]\in\mathfrak{M}$](http://images.planetmath.org:8080/cache/objects/3257/l2h/img11.png "$ \tau[G]\in\mathfrak{M}$"){kind=small}
![$ G\in\mathfrak{M}[H]$](http://images.planetmath.org:8080/cache/objects/3257/l2h/img14.png "$ G\in\mathfrak{M}[H]$"){kind=small}
![$ H\in\mathfrak{M}[G]$](http://images.planetmath.org:8080/cache/objects/3257/l2h/img15.png "$ H\in\mathfrak{M}[G]$"){kind=small}
![$ \mathfrak{M}[G]=\mathfrak{M}[H]$](http://images.planetmath.org:8080/cache/objects/3257/l2h/img16.png "$ \mathfrak{M}[G]=\mathfrak{M}[H]$"){kind=small}