Slightly different proof:
1- Fx' ∧ ∀y(Fy ⊃ y=x') Existential instantiation (P1)
2- Gy' ∧ ∀y(Gy ⊃ y=y') Existential instantiation (P2)
3- Fx' Simplification (1)
4- ∀y(Fy ⊃ y=x') Simplification (1)
5- Gy' Simplification (2)
6- ∀y(Gy ⊃ y=y') Simplification (2)
7- ¬Fx' ∨ ¬Gx' Universal instantiation (P3)
8- ¬Fy' ∨ ¬Gy' Universal instantiation (P3)
9- ¬Gx' Disjunctive syllogism (3,7)
10- ¬Fy' Disjunctive syllogism (5,8)
11- Fx' ∧ ¬Fy' Adjunction (3,10)
12- ¬(x' = y') Identity (11)
13- Fx' ∨ Gx' Addition (3)
14- Fy' ∨ Gy' Addition (5)
15- ∀y(Fy ⊃ y=x' ∨ y=y') Addition (4)
16- ∀y(Gy ⊃ y=x' ∨ y=y') Addition (6)
17- ∀y(Fy ⊃ y=x'∨y=y') ∧ ∀y(Gy ⊃ y=x' ∨ y=y') Adjunction (15,16)
18- ∀y[(Fy ⊃ y=x'∨y=y') ∧ (Gy ⊃ y=x'∨y=y')] Universal distribution (17)
19- ∀y[(¬Fy ∨(y=x'∨y=y')) ∧ (¬Gy ∨ (y=x'∨y=y'))] Implication (18)
20- ∀y[(¬Fy ∧ ¬Gy) ∨ (y=x'∨y=y')] Distribution law (19)
21- ∀y[(Fy ∨ Gy) ⊃ (y=x'∨y=y')] Implication (20)
C - ∃x∃y[ ( ( (Fx ∨ Gx) ∧ (Fy ∨ Gy) ) ∧ (x≠y) )
∧ ∀z((Fz ∨ Gz) ⊃ (z=x ∨ z=y) )] Existential Generalization (12,13,14,21)