Theorem nfiotadw 6523

Description: Deduction version of nfiotaw 6524. Version of nfiotad 6525 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2404. (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 6521	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}
2		nfv 1949	. . . 4 ⊢ Ⅎ𝑧𝜑
3		nfiotadw.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfiotadw.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5		nfvd 1950	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
6	4, 5	nfbid 1937	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
7	3, 6	nfald 2362	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
8	2, 7	nfabdw 2947	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
9	8	nfunid 4878	. 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
10	1, 9	nfcxfrd 2925	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1816  {cab 2741  Ⅎwnfc 2911  ∪ cuni 4872  ℩cio 6518

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-v 3456  df-ss 3901  df-sn 4589  df-uni 4873  df-iota 6520

This theorem is used by:  nfiotaw  6524  nfriotadw  7416