Theorem nfiotadxy 4870

Description: Deduction version of nfiotaxy 4871. (Contributed by Jim Kingdon, 21-Dec-2018.)

Hypotheses

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

Assertion

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

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfiotadxy

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 4868	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}
2		nfv 1421	. . . 4 ⊢ Ⅎ𝑧𝜑
3		nfiotadxy.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfiotadxy.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5		nfcv 2178	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
6		nfcv 2178	. . . . . . . 8 ⊢ Ⅎ𝑥𝑧
7	5, 6	nfeq 2185	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = 𝑧
8	7	a1i 9	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
9	4, 8	nfbid 1480	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
10	3, 9	nfald 1643	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
11	2, 10	nfabd 2196	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
12	11	nfunid 3587	. 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
13	1, 12	nfcxfrd 2176	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243  Ⅎwnf 1349  {cab 2026  Ⅎwnfc 2165  ∪ cuni 3580  ℩cio 4865

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-sn 3381  df-uni 3581  df-iota 4867

This theorem is referenced by:  nfiotaxy  4871  nfriotadxy  5476