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I believe other means of measuring distances have come up. I think gravitational lenses play a role, as well as certain types of variable stars that can be detectable in other galaxies. Even with doppler, one would have to posit that the speed of light has changed with time to counter the current age of the universe hypothesis. Not an impossible assumption, that, but it would still run against God's revelation that the universe is trustworthy for drawing observational inferences.

The speed of light has remained constant since it was first measured in 1676.
It is indeed theoretized that the speed of light may have slowed down over the many billions of years since the big bang, but it is highly implausible that the speed of light started out 6000 years ago at many million times its current speed, the speed then dropped off rapidly and now remains constant.

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