Theorem bj-cleljusti 31856

Description: One direction of cleljust 1985, requiring only ax-1 6-- ax-5 1827 and ax8v1 1981. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem bj-cleljusti

Step	Hyp	Ref	Expression
1		ax8v1 1981	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 → 𝑥 ∈ 𝑦))
2	1	imp 444	. 2 ⊢ ((𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)
3	2	exlimiv 1845	1 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-8 1979

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by: (None)