Theorem nfriotadxy 6037

Description: Deduction version of nfriota 6038. (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 6028	. 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		nfriotadxy.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcv 2392	. . . . . 6 ⊢ Ⅎ𝑥𝑦
4	3	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦)
5		nfriotadxy.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6	4, 5	nfeld 2408	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfriotadxy.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
8	6, 7	nfand 1621	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
9	2, 8	nfiotadw 5335	. 2 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
10	1, 9	nfcxfrd 2390	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ∧ wa 104  Ⅎwnf 1513   ∈ wcel 2209  Ⅎwnfc 2379  ℩cio 5330  ℩crio 6027

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-sn 3711  df-uni 3931  df-iota 5332  df-riota 6028

This theorem is referenced by:  nfriota  6038