On condition that 是一个固定搭配， 可用于强调某人采取行动前必须事先认可的条件，通常位于主句之后，意思是“如果”“只要”“以...为条件”等。
一、连用陈述语气
以下例句均出自词典和语法书：
I will come on condition (that) she is invited too.
如果邀请她来, 那我就来。
You can go out on condition that you wear an overcoat.
你要穿上外衣才能出去。
I’ll lend it to you on condition that you bring it back tomorrow.
我把它借给你，条件是明天你还给我。
I’ll give you the day off on condition that you work on Satiny morning.
我让你休一天假，条件是周六上午你上班。
I will do it on condition that you help me.
只要你肯帮助我，我就愿做此事。
You can take my car on condition that you drive carefully.
你可以用我的汽车，只要你小心点儿开。
We’ll let you use the room on condition that you keep it clean and tidy.
我们可以让你使用这个房间，条件是你把它保持整洁。
Shrewdly, he offered to provide financial backing on condition he became an equal partner.
他精明地主动提供财政支持，条件是他成为对等的合作伙伴。
二、连用虚拟语气
这类虚拟语气主要由 should + 动词原形 构成。如：
He can use the bicycle on condition that he should return it tomorrow.
如果他明天能还回来，就可以借用这辆自行车。——摘自《薄冰高级英语语法》
I reluctantly agreed to a postponement on condition that the sale should be completed and the boat handed over by 31st August.
只有销售事宜得以办妥，并在8月31日前交付那艘船，我才勉强同意延期。——摘自柯林斯英语语法系列《连词》