Theorem exgen

Description: Rule of existential generalization, similar to universal generalization ax-gen , but valid only if an individual exists. Its proof requires ax-6 in our axiomatization but the equality predicate does not occur in its statement. Some fundamental theorems of predicate calculus can be proven from ax-gen , ax-4 and this theorem alone, not requiring ax-7 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017) (Proof shortened by Wolf Lammen, 20-Oct-2023)

	Ref	Expression
Hypothesis	exgen.1	⊢ 𝜑
Assertion	exgen	⊢ ∃ 𝑥 𝜑

Proof

Step	Hyp	Ref	Expression
1		exgen.1	⊢ 𝜑
2		idd	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜑 ) )
3	2 1	speiv	⊢ ∃ 𝑥 𝜑