Simple logic implication question

Question

Suppose I know that $$A \iff B$$ and $$ B \implies C $$

Can I conclude that $A \land C \iff B$ ?

It seems intuitively true, but I'm not sure if there is a nice way to show it.

Answer 1

This one maybe a little more complicated. $A \iff B$ means that you can substitute $A$ into $B \implies C$ thus you have.

$A \implies C$ with this you can conclude $A \land C \iff A$

by subsituting the second $A$ with $B$ you get $A \land C \iff B$

Answer 2

$ \def\fitchproof#1#2{\begin{array}{|l}#1 \\ \hline #2\end{array}} $

$\fitchproof{ 1. A \Leftrightarrow B\\ 2. B \Rightarrow C }{ \fitchproof{ 3. A \land C }{ 4. A \quad \land Elim \ 3\\ 5. B \quad \Leftrightarrow Elim \ 1,4 }\\ \fitchproof{ 6. B} { 7. A \quad \Leftrightarrow Elim \ 1,6\\ 8. C \quad \Rightarrow Elim \ 2,6\\ 9. A \land C \quad \land Intro \ 7,8 }\\ 10. (A \land C) \Leftrightarrow B \quad \Leftrightarrow Intro \ 3-5,6-9}$