Theorem nfiotadw 6457

Description: Deduction version of nfiotaw 6458. Version of nfiotad 6459 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2376. (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 6455	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}
2		nfv 1916	. . . 4 ⊢ Ⅎ𝑧𝜑
3		nfiotadw.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfiotadw.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5		nfvd 1917	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
6	4, 5	nfbid 1904	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
7	3, 6	nfald 2333	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
8	2, 7	nfabdw 2920	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
9	8	nfunid 4856	. 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
10	1, 9	nfcxfrd 2897	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀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-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:  nfiotaw  6458  nfriotadw  7332