Theorem ac4c

Description: Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997)

	Ref	Expression
Hypothesis	ac4c.1	⊢ 𝐴 ∈ V
Assertion	ac4c	⊢ ∃ 𝑓 ∀ 𝑥 ∈ 𝐴 ( 𝑥 ≠ ∅ → ( 𝑓 ‘ 𝑥 ) ∈ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		ac4c.1	⊢ 𝐴 ∈ V
2		raleq	⊢ ( 𝑦 = 𝐴 → ( ∀ 𝑥 ∈ 𝑦 ( 𝑥 ≠ ∅ → ( 𝑓 ‘ 𝑥 ) ∈ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝑥 ≠ ∅ → ( 𝑓 ‘ 𝑥 ) ∈ 𝑥 ) ) )
3	2	exbidv	⊢ ( 𝑦 = 𝐴 → ( ∃ 𝑓 ∀ 𝑥 ∈ 𝑦 ( 𝑥 ≠ ∅ → ( 𝑓 ‘ 𝑥 ) ∈ 𝑥 ) ↔ ∃ 𝑓 ∀ 𝑥 ∈ 𝐴 ( 𝑥 ≠ ∅ → ( 𝑓 ‘ 𝑥 ) ∈ 𝑥 ) ) )
4		ac4	⊢ ∃ 𝑓 ∀ 𝑥 ∈ 𝑦 ( 𝑥 ≠ ∅ → ( 𝑓 ‘ 𝑥 ) ∈ 𝑥 )
5	1 3 4	vtocl	⊢ ∃ 𝑓 ∀ 𝑥 ∈ 𝐴 ( 𝑥 ≠ ∅ → ( 𝑓 ‘ 𝑥 ) ∈ 𝑥 )