Theorem nfiotadw

Description: Deduction version of nfiotaw . Version of nfiotad with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 18-Feb-2013) (Revised by Gino Giotto, 26-Jan-2024)

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

Proof

Step	Hyp	Ref	Expression
1		nfiotadw.1	⊢ Ⅎ 𝑦 𝜑
2		nfiotadw.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
3		dfiota2	⊢ ( ℩ 𝑦 𝜓 ) = ∪ { 𝑧 ∣ ∀ 𝑦 ( 𝜓 ↔ 𝑦 = 𝑧 ) }
4		nfv	⊢ Ⅎ 𝑧 𝜑
5		nfvd	⊢ ( 𝜑 → Ⅎ 𝑥 𝑦 = 𝑧 )
6	2 5	nfbid	⊢ ( 𝜑 → Ⅎ 𝑥 ( 𝜓 ↔ 𝑦 = 𝑧 ) )
7	1 6	nfald	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 ( 𝜓 ↔ 𝑦 = 𝑧 ) )
8	4 7	nfabdw	⊢ ( 𝜑 → Ⅎ 𝑥 { 𝑧 ∣ ∀ 𝑦 ( 𝜓 ↔ 𝑦 = 𝑧 ) } )
9	8	nfunid	⊢ ( 𝜑 → Ⅎ 𝑥 ∪ { 𝑧 ∣ ∀ 𝑦 ( 𝜓 ↔ 𝑦 = 𝑧 ) } )
10	3 9	nfcxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 ( ℩ 𝑦 𝜓 ) )