Khedivate of Egypt conquers Ethiopia

In Egyptian–Ethiopian War the Khedivate of Egypt Isma'il Pasha invaded the Ethiopian Empire under Yohannes IV in 1874. But the Egyptian army was ambushed by the Ethiopians at Gundet in a mountain pass . Later at Gura Egyptian demolition of trenches despite advice of foreign commander lead to their defeat at the hand of Ethiopians. POD is that Egyptians counter the ambush at Gundet and destroy the Ethiopian force. In later battles Ethiopians are defeated by clever use trenches by Egyptians.

 
It's not impossible that the Egyptians pull off some ultra level diplomacy and play the various ethnic groups against each other. Utilising clan feuds, backing local powerbrokers and playing off local politics to hold onto the region.

Also the geography is somewhat in favour as they can use the River Nile to send troops and resources into Northern Ethiopia. Purchasing modern European weaponry for a few elite regiments would make holding onto territory there very possible

However the thing that I'm not certain about is money; I don't know what the Egyptian economy is like in this time period; could they afford this adventure is the main question. Also would it make enough returns to justify itself would be another.
 
Also the geography is somewhat in favour as they can use the River Nile to send troops and resources into Northern Ethiopia.
There are multiple cataracts (rapids) that they have to deal with in between, though:

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Greece, in particular, given how Greece has had positive relations with the Christian Ethiopians throughout history, would likely have frosty or even negative relations with Egypt.
 
Greece, in particular, given how Greece has had positive relations with the Christian Ethiopians throughout history, would likely have frosty or even negative relations with Egypt.

Greece couldn't do anything else as rant anyway. Brits and massive very angry Christian population would are bigger worry to Egypt.
 
Even if the Egyptians overturn the ambush, that doesn't mean they can defeat the Ethiopian army. I actually can't think of any point after breaking away from the Ottomans the Egyptian army being capable of invading and defeating Ethiopia.
 
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