Vamos supor que $[\![ \varphi\implies \psi ]\!] = 1$:
$$1 = [\![\varphi \implies \psi]\!]=[\![\neg \varphi \vee \psi]\!] = -[\![\varphi]\!] + [\![\psi]\!]$$
Vamos supor que $[\![\varphi]\!] \leq [\![\psi]\!]$
$$1 = [\![\varphi]\!]\implies [\![\psi]\!] = -[\![\varphi]\!] + [\![\psi]\!] = [\![\neg\varphi\vee\psi]\!] = [\![ \varphi\implies \psi ]\!]$$ $\square$