Theorem nfiotadw 6394

Description: Deduction version of nfiotaw 6395. Version of nfiotad 6396 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 18-Feb-2013.) (Revised by Gino Giotto, 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 6392	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}
2		nfv 1917	. . . 4 ⊢ Ⅎ𝑧𝜑
3		nfiotadw.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfiotadw.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5		nfvd 1918	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
6	4, 5	nfbid 1905	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
7	3, 6	nfald 2322	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
8	2, 7	nfabdw 2930	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
9	8	nfunid 4845	. 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
10	1, 9	nfcxfrd 2906	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  Ⅎwnf 1786  {cab 2715  Ⅎwnfc 2887  ∪ cuni 4839  ℩cio 6389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-v 3434  df-in 3894  df-ss 3904  df-sn 4562  df-uni 4840  df-iota 6391

This theorem is referenced by:  nfiotaw  6395  nfriotadw  7240