Theorem ac6sf2

Description: Alternate version of ac6 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008) (Revised by Thierry Arnoux, 17-May-2020)

	Ref	Expression
Hypotheses	ac6sf2.y	⊢ Ⅎ 𝑦 𝐵
	ac6sf2.1	⊢ Ⅎ 𝑦 𝜓
	ac6sf2.2	⊢ 𝐴 ∈ V
	ac6sf2.3	⊢ ( 𝑦 = ( 𝑓 ‘ 𝑥 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	ac6sf2	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑓 ( 𝑓 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ac6sf2.y	⊢ Ⅎ 𝑦 𝐵
2		ac6sf2.1	⊢ Ⅎ 𝑦 𝜓
3		ac6sf2.2	⊢ 𝐴 ∈ V
4		ac6sf2.3	⊢ ( 𝑦 = ( 𝑓 ‘ 𝑥 ) → ( 𝜑 ↔ 𝜓 ) )
5		nfcv	⊢ Ⅎ 𝑧 𝐵
6		nfv	⊢ Ⅎ 𝑧 𝜑
7		nfs1v	⊢ Ⅎ 𝑦 [ 𝑧 / 𝑦 ] 𝜑
8		sbequ12	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜑 ) )
9	1 5 6 7 8	cbvrexfw	⊢ ( ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑧 ∈ 𝐵 [ 𝑧 / 𝑦 ] 𝜑 )
10	9	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐵 [ 𝑧 / 𝑦 ] 𝜑 )
11	2 4	sbhypf	⊢ ( 𝑧 = ( 𝑓 ‘ 𝑥 ) → ( [ 𝑧 / 𝑦 ] 𝜑 ↔ 𝜓 ) )
12	3 11	ac6s	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐵 [ 𝑧 / 𝑦 ] 𝜑 → ∃ 𝑓 ( 𝑓 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) )
13	10 12	sylbi	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑓 ( 𝑓 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) )