Theorem ac6sf

Description: Version of ac6 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008)

	Ref	Expression
Hypotheses	ac6sf.1	⊢ Ⅎ 𝑦 𝜓
	ac6sf.2	⊢ 𝐴 ∈ V
	ac6sf.3	⊢ ( 𝑦 = ( 𝑓 ‘ 𝑥 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	ac6sf	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑓 ( 𝑓 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ac6sf.1	⊢ Ⅎ 𝑦 𝜓
2		ac6sf.2	⊢ 𝐴 ∈ V
3		ac6sf.3	⊢ ( 𝑦 = ( 𝑓 ‘ 𝑥 ) → ( 𝜑 ↔ 𝜓 ) )
4		cbvrexsvw	⊢ ( ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑧 ∈ 𝐵 [ 𝑧 / 𝑦 ] 𝜑 )
5	4	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐵 [ 𝑧 / 𝑦 ] 𝜑 )
6	1 3	sbhypf	⊢ ( 𝑧 = ( 𝑓 ‘ 𝑥 ) → ( [ 𝑧 / 𝑦 ] 𝜑 ↔ 𝜓 ) )
7	2 6	ac6s	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐵 [ 𝑧 / 𝑦 ] 𝜑 → ∃ 𝑓 ( 𝑓 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) )
8	5 7	sylbi	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑓 ( 𝑓 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) )