Theorem nfriotadxy 5931

Description: Deduction version of nfriota 5932. (Contributed by Jim Kingdon, 12-Jan-2019.)

Hypotheses

Ref	Expression
nfriotadxy.1	⊢ Ⅎ𝑦𝜑
nfriotadxy.2	⊢ (𝜑 → Ⅎ𝑥𝜓)
nfriotadxy.3	⊢ (𝜑 → Ⅎ𝑥𝐴)

Assertion

Ref	Expression
nfriotadxy	⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfriotadxy

Step	Hyp	Ref	Expression
1		df-riota 5922	. 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		nfriotadxy.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcv 2350	. . . . . 6 ⊢ Ⅎ𝑥𝑦
4	3	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦)
5		nfriotadxy.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6	4, 5	nfeld 2366	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfriotadxy.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
8	6, 7	nfand 1592	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
9	2, 8	nfiotadw 5254	. 2 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
10	1, 9	nfcxfrd 2348	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ∧ wa 104  Ⅎwnf 1484   ∈ wcel 2178  Ⅎwnfc 2337  ℩cio 5249  ℩crio 5921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-sn 3649  df-uni 3865  df-iota 5251  df-riota 5922

This theorem is referenced by:  nfriota  5932