Proof of Strong Duality Via Farkas Lemma
The following is based on the book "Combinatorial Optimization" by Papadimitriou and Steiglitz (chapter 3)

On LP duality, they first state a theorem that when a LP has a feasible solution, the dual also has one and on optimality their costs are equal (Theorem 3.1 in the book, based on the simplex algorithm)

This follows from the fact that the dual cost is always dominated by the primal cost.

i.e let $x$ and $\pi$ the solutions to the primal and dual respectively, then (i.e $c'$ is transpose vector of $c$)

$$c'x \ge \pi'Ax \ge \pi'b$$

So if the primal has a feasible solution so does the dual and on optimality they (the costs) are equal.

Then the Farkas Lemma (theorem 3.5 in the book) is proved based on this theorem.

But one can reverse the proof of Farkas Lemma in the book and prove Th. 3.1 from Farkas Lemma (Th. 3.5)

A sketch:

Farkas' Lemma states that (as stated in book):


  Given vectors $a_i \in R^n$, $i = 1,..m$ and another vector $c \in R^n$,
  
  iff $y'a_i \ge 0, \forall i \implies y'c \ge 0$, then $c \in C(a_i)$
  (i.e $c$ is in the cone generated by $a_i$)


Consider a LP:

$$\min{c'y}$$
$$a_i'y \ge 0, i =1..m$$
$$y \ne 0$$

The LP is feasible and bounded because $y = 0$ is a feasible point and $c'y \ge 0$ (by the lemma)

$c$ can be expressed as linear combination of the $a_i$'s using a vector $\pi$, where $\pi_i \ge 0$, i.e

$$c = \sum_1^m{\pi_i a_i}$$

Constructing the dual of the LP as:

$$A_j = col(a_{ij}, i=1..m) \in R^m$$
$$a_i = col(a_{ij}, j=1..n) \in R^n$$

gives the dual LP:

$$\max{0}$$
$$\pi'A_j = c_j, j = 1..n$$
$$\pi \ge 0$$

which has a feasible solution ($\pi$) and equal cost at optimality (strong duality theorem)