Theorem nfiotad 6459

Description: Deduction version of nfiota 6460. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker nfiotadw 6457 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 6455	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}
2		nfv 1916	. . . 4 ⊢ Ⅎ𝑧𝜑
3		nfiotad.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfiotad.2	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
6		nfeqf1 2383	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
7	6	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
8	5, 7	nfbid 1904	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
9	3, 8	nfald2 2449	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
10	2, 9	nfabd 2921	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
11	10	nfunid 4856	. 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
12	1, 11	nfcxfrd 2897	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540  Ⅎwnf 1785  {cab 2714  Ⅎwnfc 2883  ∪ cuni 4850  ℩cio 6452

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-13 2376  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-v 3431  df-ss 3906  df-sn 4568  df-uni 4851  df-iota 6454

This theorem is referenced by:  nfiota  6460  nfriotad  7335