There are two or more ways to define an Erdos-Rényi random graph. Let consider the following two:

1) $G_n=(V_n,E_n)$ with vertex set $V_n=(1,\dots,n)$ and edge set $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ \epsilon_{ij}=1)$, where $(\epsilon_{ij})_{ij}$ are i.i.d. random variables with distribution Bernoulli($2c/(n-1)$).

2) $G_n^*=(V_n,E_n^*)$ with (multi)-edge set $E_n^*=(i_1j_1,\ i_2j_2,\ \dots,\ i_mj_m)$ where $m=[cn]$ and $(i_s)_s\cup(j_s)_s$ are i.i.d. random vertices uniformly chosen in $V_n$

Notice that $G_n^*$ is in general a multigraph, i.e. it admits self-loops and multi-edges, whilest $G_n$ is a simple graph. Notice that $G_n^*$ has a deterministic number $m=[cn]$ of edges, while $G_n$ has a random number of edges whose expected value is $cn$.

I read that $$G_n\ \overset{d}{=}\ G_n^*\ |\ ''G_n^*\ \text{is simple}''$$ and that $$\mathbb{P}(G_n^*\ \text{is simple})\geq\delta>0\ \ \forall n\in\mathbb{N}.$$

Could you help me to prove this two fact, or tell me where I can find a proof?

**Edit.** The two distributions can't be equal at fixed $n$ (see Greg Martin's answer). But I hope they're asymptotically equivalent.

If this can help, I think the law of $G_n^*$ conditionally on ($G_n^*$ simple) is the uniform distribution on the set $\mathcal{G}_{n,m}$ of simple graphs with vertex set $V_n$ and $m$ edges.