Quantum states' inner product is time independent

$⟨\varphi (t)\mid \psi (t)⟩$ is independent of $t$ for 2 solutions $\psi ,\varphi$ of the same equation ( Schrodinger Equation ), where

1. $\psi ,\varphi$ are time parameterized Quantum state solutions to Schrodinger Equation
2. $⟨\cdot ,\cdot ⟩:H\times H\to \mathbb{C}$ is an Inner Product on the Quantum Hilbert Space
3. $t\in {\mathbb{R}}^{+}$ denotes time.

$$\begin{array}{rll}\frac{d}{dt}⟨\varphi ,\psi ⟩& =⟨\frac{d}{dt}\varphi ,\psi ⟩+⟨\varphi ,\frac{d}{dt}\psi ⟩& \text{Using the product rule for derivatives}\\ & =⟨\frac{1}{i\hslash }\hat{H}\varphi ,\psi ⟩+⟨\varphi ,\frac{1}{i\hslash }\hat{H}\psi ⟩& \text{By Schrodinger’s Equation}\\ & =-\frac{1}{i\hslash }⟨\hat{H}\varphi ,\psi ⟩+\frac{1}{i\hslash }⟨\varphi ,\hat{H}\psi ⟩& \text{Using the linearity of the Inner Product}\\ & =-\frac{1}{i\hslash }⟨\hat{H}\varphi ,\psi ⟩+\frac{1}{i\hslash }⟨\hat{H}\varphi ,\psi ⟩& \text{Since H^ is self adjoint}\\ & =0& \end{array}$$

Hence $\frac{d}{dt}⟨\varphi ,\psi ⟩=0$, so the Inner Product $⟨\varphi ,\psi ⟩$ is time-independent