but peskym... "His can't, on the whole."... the fact that men put up with this situation simply perpetuates the "woman's job = flexible, low priority, emergency childcare expendable" and "dad's job = not flexible, higher priority, can't do emergency childcare" myth.
the fact is that, unless your husband works in an environment where there is literally no one else that can do his job at the very minute he's not there (say, surgeon, in a courtroom, or IT troubleshooter - something where the expertise is needed then and there, no questions), it's simply poorly structured work environments and bad management which have brainwashed couples like you into thinking "can't possibly get DH to do emergency childcare".
maybe it genuinely IS the case for you, you haven't said what you both actually do, but the older i get the more i see poor working environments are the reason why men feel they "can't" take more than a week off for paternity leave, "can't" go on the family holiday during summer without a laptop... the sooner men demand more of an involvement in their child's upbringing (emergency childcare is but one component) then easier it will be for working mums to feel less pressure
i hope the situation improves for you over time 
but my DH has been where yours is - he is an IT programmer, and when we both felt he was working too much (basically working 80hrs a week some weeks due to being on call, never seeing kids before bed, totally uninvolved despite wanting to be)... he wanted to cut back to 3 or 4 days. his boss wouldn't even entertain it.
so i did more hours and DH quit. found an employer who WOULD understand that dads need to be involved in care of children too, and went 3 days a week for two years. he was lucky in that he found one.
but until the majority of employers are forced to see that their family unfriendly ways of working are losing them talent, training and the people they want to keep - where is the incentive for them to become more flexible, such as for your DH?
sorry but shit employers strike a raw nerve with me these days!!