Theorem nfiotadxy 4978

Description: Deduction version of nfiotaxy 4979. (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 4976	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}
2		nfv 1466	. . . 4 ⊢ Ⅎ𝑧𝜑
3		nfiotadxy.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4		nfiotadxy.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5		nfcv 2228	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
6		nfcv 2228	. . . . . . . 8 ⊢ Ⅎ𝑥𝑧
7	5, 6	nfeq 2236	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = 𝑧
8	7	a1i 9	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
9	4, 8	nfbid 1525	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
10	3, 9	nfald 1690	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
11	2, 10	nfabd 2247	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
12	11	nfunid 3658	. 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)})
13	1, 12	nfcxfrd 2226	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1287   = wceq 1289  Ⅎwnf 1394  {cab 2074  Ⅎwnfc 2215  ∪ cuni 3651  ℩cio 4973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-sn 3450  df-uni 3652  df-iota 4975

This theorem is referenced by:  nfiotaxy  4979  nfriotadxy  5608