Theorem axac3

Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 as an axiom. (Contributed by Mario Carneiro, 6-May-2015) (Revised by NM, 20-Dec-2016) (Proof modification is discouraged.)

	Ref	Expression
Assertion	axac3	⊢ CHOICE

Proof

Step	Hyp	Ref	Expression
1		ax-ac2	⊢ ∃ 𝑦 ∀ 𝑧 ∃ 𝑤 ∀ 𝑣 ( ( 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑦 → ( ( 𝑤 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑤 ) ∧ 𝑧 ∈ 𝑤 ) ) ) ∨ ( ¬ 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑥 → ( ( 𝑤 ∈ 𝑧 ∧ 𝑤 ∈ 𝑦 ) ∧ ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) → 𝑣 = 𝑤 ) ) ) ) )
2	1	ax-gen	⊢ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ∃ 𝑤 ∀ 𝑣 ( ( 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑦 → ( ( 𝑤 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑤 ) ∧ 𝑧 ∈ 𝑤 ) ) ) ∨ ( ¬ 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑥 → ( ( 𝑤 ∈ 𝑧 ∧ 𝑤 ∈ 𝑦 ) ∧ ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) → 𝑣 = 𝑤 ) ) ) ) )
3		dfackm	⊢ ( CHOICE ↔ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ∃ 𝑤 ∀ 𝑣 ( ( 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑦 → ( ( 𝑤 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑤 ) ∧ 𝑧 ∈ 𝑤 ) ) ) ∨ ( ¬ 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑥 → ( ( 𝑤 ∈ 𝑧 ∧ 𝑤 ∈ 𝑦 ) ∧ ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) → 𝑣 = 𝑤 ) ) ) ) ) )
4	2 3	mpbir	⊢ CHOICE