Theorem nfiotadw 5091

Description: Bound-variable hypothesis builder for the ℩ class. (Contributed by Jim Kingdon, 21-Dec-2018.)

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 5089	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}
2		nfv 1508	. . . 4 ⊢ Ⅎ𝑧𝜑
3		nfiotadw.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfiotadw.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5		nfcv 2281	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
6		nfcv 2281	. . . . . . . 8 ⊢ Ⅎ𝑥𝑧
7	5, 6	nfeq 2289	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = 𝑧
8	7	a1i 9	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
9	4, 8	nfbid 1567	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
10	3, 9	nfald 1733	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
11	2, 10	nfabd 2300	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
12	11	nfunid 3743	. 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
13	1, 12	nfcxfrd 2279	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329   = wceq 1331  Ⅎwnf 1436  {cab 2125  Ⅎwnfc 2268  ∪ cuni 3736  ℩cio 5086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-sn 3533  df-uni 3737  df-iota 5088

This theorem is referenced by:  nfiotaw  5092  nfriotadxy  5738