Theorem nfiotad

Description: Deduction version of nfiota . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfiotadw when possible. (Contributed by NM, 18-Feb-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfiotad.1	⊢ Ⅎ 𝑦 𝜑
	nfiotad.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion	nfiotad	⊢ ( 𝜑 → Ⅎ 𝑥 ( ℩ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		nfiotad.1	⊢ Ⅎ 𝑦 𝜑
2		nfiotad.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
3		dfiota2	⊢ ( ℩ 𝑦 𝜓 ) = ∪ { 𝑧 ∣ ∀ 𝑦 ( 𝜓 ↔ 𝑦 = 𝑧 ) }
4		nfv	⊢ Ⅎ 𝑧 𝜑
5	2	adantr	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 )
6		nfeqf1	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑦 = 𝑧 )
7	6	adantl	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝑦 = 𝑧 )
8	5 7	nfbid	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 ( 𝜓 ↔ 𝑦 = 𝑧 ) )
9	1 8	nfald2	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 ( 𝜓 ↔ 𝑦 = 𝑧 ) )
10	4 9	nfabd	⊢ ( 𝜑 → Ⅎ 𝑥 { 𝑧 ∣ ∀ 𝑦 ( 𝜓 ↔ 𝑦 = 𝑧 ) } )
11	10	nfunid	⊢ ( 𝜑 → Ⅎ 𝑥 ∪ { 𝑧 ∣ ∀ 𝑦 ( 𝜓 ↔ 𝑦 = 𝑧 ) } )
12	3 11	nfcxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 ( ℩ 𝑦 𝜓 ) )