Theorem bj-cleljusti 37164

Description: One direction of cleljust 2154, requiring only ax-1 6-- ax-5 1933 and ax8v1 2149. (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 2149	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 → 𝑥 ∈ 𝑦))
2	1	imp 411	. 2 ⊢ ((𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)
3	2	exlimiv 1953	1 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-8 2147

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by: (None)