Theorem nfriotadxy 5731

Description: Deduction version of nfriota 5732. (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 5723	. 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		nfriotadxy.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcv 2279	. . . . . 6 ⊢ Ⅎ𝑥𝑦
4	3	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦)
5		nfriotadxy.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6	4, 5	nfeld 2295	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfriotadxy.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
8	6, 7	nfand 1547	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
9	2, 8	nfiotadw 5086	. 2 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
10	1, 9	nfcxfrd 2277	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ∧ wa 103  Ⅎwnf 1436   ∈ wcel 1480  Ⅎwnfc 2266  ℩cio 5081  ℩crio 5722

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 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-sn 3528  df-uni 3732  df-iota 5083  df-riota 5723

This theorem is referenced by:  nfriota  5732