Theorem nfiotad 6499

Description: Deduction version of nfiota 6500. Usage of this theorem is discouraged because it depends on ax-13 2401. Use the weaker nfiotadw 6497 when possible. (Contributed by NM, 18-Feb-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfiotad.1	⊢ Ⅎ𝑦𝜑
nfiotad.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfiotad	⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))

Proof of Theorem nfiotad

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 6495	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}
2		nfv 1946	. . . 4 ⊢ Ⅎ𝑧𝜑
3		nfiotad.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfiotad.2	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
6		nfeqf1 2408	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
7	6	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
8	5, 7	nfbid 1934	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
9	3, 8	nfald2 2474	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
10	2, 9	nfabd 2944	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
11	10	nfunid 4872	. 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
12	1, 11	nfcxfrd 2921	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567  Ⅎwnf 1815  {cab 2738  Ⅎwnfc 2907  ∪ cuni 4866  ℩cio 6492

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3896  df-sn 4585  df-uni 4867  df-iota 6494

This theorem is used by:  nfiota  6500  nfriotad  7389