de-humidifying and heating

Had a chat yesterday with a customer. we got round to domestic heating (for some strage reason) He said thay he'd been told that by having a dehum running during winter this would cut his fuel bills. The reasoning was that dry air is easier to heat than moist. I suggested the running of the dehum would counter the savings but he was told no over all it is cheaper overall to use both

I told him his main problem was that he had a huge house and no loft insulation, cavity wall insualtion or double glazing and to get that sorted first

Still that aside any thoughts and proof either way would be welcome

From 12°C 90% to 24°C 42% (sensible heating only), the difference in enthalpy is 12 kJ/Kg.

From 12°C 30% to 24°C 15% (sensible heating only), the difference in enthalpy is the same.

Therefore there is no difference whether you heat moist or dry air.