Very few things are directly deductible from federal income tax. Most deductions are from the taxable income.
So if you pay X in state income taxes, you'd deduct X from your overall federally taxed income. You end up only saving your federal tax rate times X, not X in federal taxes. So if you're taxed at 13% state and 35% federally, you'd pay 13% to the state. Then you'd still pay (barring other deductions) your federal taxes on 87% of your income.
Federal taxes don't work as a bare percentage, though. You pay a lower percentage on everything up to some number, then progressively higher percentages on brackets of your income up to your maximum tax rate. https://taxfoundation.org/2017-tax-brackets/
So if you pay X in state income taxes, you'd deduct X from your overall federally taxed income. You end up only saving your federal tax rate times X, not X in federal taxes. So if you're taxed at 13% state and 35% federally, you'd pay 13% to the state. Then you'd still pay (barring other deductions) your federal taxes on 87% of your income.
Federal taxes don't work as a bare percentage, though. You pay a lower percentage on everything up to some number, then progressively higher percentages on brackets of your income up to your maximum tax rate. https://taxfoundation.org/2017-tax-brackets/