Theorem nfiotadw 6480

Description: Deduction version of nfiotaw 6481. Version of nfiotad 6482 with a disjoint variable condition, which does not require ax-13 2403. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2403. (Revised by GG, 26-Jan-2024.)

Hypotheses

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

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem nfiotadw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 6478	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}
2		nfv 1934	. . . 4 ⊢ Ⅎ𝑧𝜑
3		nfiotadw.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfiotadw.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5		nfvd 1935	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
6	4, 5	nfbid 1922	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
7	3, 6	nfald 2360	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
8	2, 7	nfabdw 2945	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
9	8	nfunid 4871	. 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
10	1, 9	nfcxfrd 2923	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1558  Ⅎwnf 1803  {cab 2740  Ⅎwnfc 2909  ∪ cuni 4865  ℩cio 6475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-v 3456  df-ss 3921  df-sn 4583  df-uni 4866  df-iota 6477

This theorem is referenced by:  nfiotaw  6481  nfriotadw  7361