Theorem nfiotad 6489

Description: Deduction version of nfiota 6490. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker nfiotadw 6487 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 6485	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}
2		nfv 1917	. . . 4 ⊢ Ⅎ𝑧𝜑
3		nfiotad.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfiotad.2	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
6		nfeqf1 2377	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
7	6	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
8	5, 7	nfbid 1905	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
9	3, 8	nfald2 2443	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
10	2, 9	nfabd 2927	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
11	10	nfunid 4907	. 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
12	1, 11	nfcxfrd 2901	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539  Ⅎwnf 1785  {cab 2708  Ⅎwnfc 2882  ∪ cuni 4901  ℩cio 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2370  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-v 3475  df-in 3951  df-ss 3961  df-sn 4623  df-uni 4902  df-iota 6484

This theorem is referenced by:  nfiota  6490  nfriotad  7361