Theorem bj-cleljusti

Description: One direction of cleljust , requiring only ax-1 -- ax-5 and ax8v1 . (Contributed by BJ, 31-Dec-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-cleljusti	⊢ ( ∃ 𝑧 ( 𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦 ) → 𝑥 ∈ 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		ax8v1	⊢ ( 𝑧 = 𝑥 → ( 𝑧 ∈ 𝑦 → 𝑥 ∈ 𝑦 ) )
2	1	imp	⊢ ( ( 𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦 ) → 𝑥 ∈ 𝑦 )
3	2	exlimiv	⊢ ( ∃ 𝑧 ( 𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦 ) → 𝑥 ∈ 𝑦 )