Theorem ac6s3

Description: Generalization of the Axiom of Choice to classes. Theorem 10.46 of TakeutiZaring p. 97. (Contributed by NM, 3-Nov-2004)

	Ref	Expression
Hypotheses	ac6s.1	⊢ 𝐴 ∈ V
	ac6s.2	⊢ ( 𝑦 = ( 𝑓 ‘ 𝑥 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	ac6s3	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 𝜑 → ∃ 𝑓 ∀ 𝑥 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ac6s.1	⊢ 𝐴 ∈ V
2		ac6s.2	⊢ ( 𝑦 = ( 𝑓 ‘ 𝑥 ) → ( 𝜑 ↔ 𝜓 ) )
3	1 2	ac6s2	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 𝜑 → ∃ 𝑓 ( 𝑓 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) )
4		exsimpr	⊢ ( ∃ 𝑓 ( 𝑓 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑓 ∀ 𝑥 ∈ 𝐴 𝜓 )
5	3 4	syl	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 𝜑 → ∃ 𝑓 ∀ 𝑥 ∈ 𝐴 𝜓 )