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Not really a good answer. I have the same issue. My phone is connected to wi-fi, but when I turn on wi-fi calling I don't get the wi-fi just after T-Mobile. So, this question isn't satisfactorily answered despite its brevity. Perhaps someone else can help.
I use an iPhone 5, and that's not true. That's pure myth.According to Apple (at least on the iPhone 5) the device will chose the faster connection. So, if your Wifi connection isn't as fast as your LTE, the device will show and use LTE instead of your Wifi (even though you are connected). If this is the case (that your LTE is faster than your Wifi) I would look into what is going on with your Wifi connection because it should be quite a bit faster than LTE (generally). Hope this helps.
The fact that this issue is ongoing for several years now (just saw it again today on my iPhone 6s) speaks to poor software quality on Apple's part. No such bug should hang around that long. No customer should have to go to settings to confirm their wireless connection.
Wouldn't this simply be the Wi-Fi Assist feature? It was introduced in iOS 9. Cell data will automatically be used when the WiFi connection is poor. If you're worried...turn the feature off.
The attached screenshot is a better representation of the issue at hand. On a v10.3.1 iphone 6s, the wifi is on, the setting screen indicates it's connected to the home wifi (Sixineight5), other devices are connected and surfing to the same wifi, except this phone continues to display LTE instead of wifi symbol at the header ribbon, therefore it's not specific if we're burning network data or iphone is experiencing a problem properly connected to wifi.
I'd propose running a speedtest.net, to see if the network is hosted by verizon (my cellular) or comcast (my home provider), but this it isn't an actual solution to why the wifi icon isn't appearing. Any insights from the forum?
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