Theorem nfiotad 6498

Description: Deduction version of nfiota 6499. Usage of this theorem is discouraged because it depends on ax-13 2403. Use the weaker nfiotadw 6496 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 6494	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}
2		nfv 1947	. . . 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 2410	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
7	6	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
8	5, 7	nfbid 1935	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
9	3, 8	nfald2 2476	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
10	2, 9	nfabd 2946	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
11	10	nfunid 4876	. 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
12	1, 11	nfcxfrd 2923	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  Ⅎwnf 1816  {cab 2740  Ⅎwnfc 2909  ∪ cuni 4870  ℩cio 6491

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-13 2403  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-v 3455  df-ss 3919  df-sn 4588  df-uni 4871  df-iota 6493

This theorem is used by:  nfiota  6499  nfriotad  7385