Theorem nfiotad 6472

Description: Deduction version of nfiota 6473. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker nfiotadw 6470 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 6468	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}
2		nfv 1914	. . . 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 2378	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
7	6	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
8	5, 7	nfbid 1902	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
9	3, 8	nfald2 2444	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
10	2, 9	nfabd 2915	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
11	10	nfunid 4880	. 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
12	1, 11	nfcxfrd 2891	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  Ⅎwnf 1783  {cab 2708  Ⅎwnfc 2877  ∪ cuni 4874  ℩cio 6465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2371  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-v 3452  df-ss 3934  df-sn 4593  df-uni 4875  df-iota 6467

This theorem is referenced by:  nfiota  6473  nfriotad  7358