For two loci, Mendelian inheritance requires two separate questions:

1. does each allele pair segregate?
2. are the allele choices at the two loci independent?

Consider an individual with genotype

$$AaBb.$$

If the $A/a$ and $B/b$ loci are on different chromosome pairs, the orientation of the $A/a$ homolog pair at meiosis I is independent of the orientation of the $B/b$ pair. The four gamete genotypes are therefore

$$AB,\quad Ab,\quad aB,\quad ab,$$

with probability

$$\frac14$$

each under ideal independent assortment.

This follows from multiplying independent allele probabilities:

$$P(AB)=P(A)P(B)=\frac12\cdot\frac12=\frac14.$$

A dihybrid cross can be decomposed into two monohybrid crosses

For

$$AaBb\times AaBb,$$

the probability of genotype $aa$ at the first locus is $1/4$, and the probability of genotype $bb$ at the second locus is $1/4$.

If the loci assort independently,

$$P(aabb)=\frac14\cdot\frac14=\frac1{16}.$$

Under complete dominance at both loci, the probability of showing both recessive phenotypes is therefore also $1/16$.

Similarly,

$$P(A_B_)=\frac34\cdot\frac34=\frac9{16},$$

where $A_$ means either $AA$ or $Aa$, and $B_$ means either $BB$ or $Bb$.

The familiar dihybrid phenotype ratio

$$9:3:3:1$$

is therefore the product structure of two independent $3:1$ monohybrid phenotype distributions.

Independent assortment is conditional, not universal

Genes on different chromosomes normally inherit the independence of their chromosome pairs. Genes on the same chromosome can fail to assort independently because they are physically linked, although crossover can separate linked allele combinations.

Thus "Mendel's law of independent assortment" should not be read as saying that every pair of genes in a genome is independent. It is an accurate model for loci whose transmission events are effectively independent.

The key bridge from cell biology to genetics is:

independent homolog-pair orientation in meiosis
                 ↓
independent allele combinations in gametes
                 ↓
product rule for multilocus inheritance