Theorem nfriotadxy 5963

Description: Deduction version of nfriota 5964. (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 5954	. 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		nfriotadxy.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcv 2372	. . . . . 6 ⊢ Ⅎ𝑥𝑦
4	3	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦)
5		nfriotadxy.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6	4, 5	nfeld 2388	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfriotadxy.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
8	6, 7	nfand 1614	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
9	2, 8	nfiotadw 5281	. 2 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
10	1, 9	nfcxfrd 2370	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ∧ wa 104  Ⅎwnf 1506   ∈ wcel 2200  Ⅎwnfc 2359  ℩cio 5276  ℩crio 5953

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-sn 3672  df-uni 3889  df-iota 5278  df-riota 5954

This theorem is referenced by:  nfriota  5964