Theorem nfriotadw 7383

Description: Deduction version of nfriota 7388 with a disjoint variable condition, which contrary to nfriotad 7387 does not require ax-13 2365. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2365. (Revised by GG, 26-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
nfriotadw	⊢ (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 nfriotadw

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-riota 7375	. 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		nfriotadw.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
3		nfnaew 2137	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
4	2, 3	nfan 1894	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
5		nfcvd 2892	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
6	5	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦)
7		nfriotadw.3	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝐴)
8	7	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴)
9	6, 8	nfeld 2903	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴)
10		nfriotadw.2	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜓)
11	10	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
12	9, 11	nfand 1892	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
13	4, 12	nfiotadw 6504	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
14	13	ex 411	. . 3 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))))
15		nfiota1 6503	. . . 4 ⊢ Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
16		biidd 261	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))))
17	16	drnf1v 2363	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ Ⅎ𝑦 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))))
18	17	albidv 1915	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑤Ⅎ𝑥 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))))
19		df-nfc 2877	. . . . 5 ⊢ (Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ ∀𝑤Ⅎ𝑥 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
20		df-nfc 2877	. . . . 5 ⊢ (Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
21	18, 19, 20	3bitr4g 313	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))))
22	15, 21	mpbiri 257	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
23	14, 22	pm2.61d2 181	. 2 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
24	1, 23	nfcxfrd 2890	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394  ∀wal 1531  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2875  ℩cio 6499  ℩crio 7374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-v 3463  df-ss 3961  df-sn 4631  df-uni 4910  df-iota 6501  df-riota 7375

This theorem is referenced by:  nfriota  7388