Theorem nfriotadw 7375

Description: Deduction version of nfriota 7379 with a disjoint variable condition, which contrary to nfriotad 7378 does not require ax-13 2402. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2402. (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 7367	. 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		nfriotadw.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
3		nfnaew 2182	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
4	2, 3	nfan 1927	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
5		nfcvd 2924	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
6	5	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦)
7		nfriotadw.3	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝐴)
8	7	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴)
9	6, 8	nfeld 2934	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴)
10		nfriotadw.2	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜓)
11	10	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
12	9, 11	nfand 1925	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
13	4, 12	nfiotadw 6495	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
14	13	ex 417	. . 3 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))))
15		nfiota1 6494	. . . 4 ⊢ Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
16		biidd 265	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))))
17	16	drnf1v 2401	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ Ⅎ𝑦 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))))
18	17	albidv 1948	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑤Ⅎ𝑥 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))))
19		df-nfc 2910	. . . . 5 ⊢ (Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ ∀𝑤Ⅎ𝑥 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
20		df-nfc 2910	. . . . 5 ⊢ (Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
21	18, 19, 20	3bitr4g 317	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))))
22	15, 21	mpbiri 261	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
23	14, 22	pm2.61d2 183	. 2 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
24	1, 23	nfcxfrd 2922	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1566  Ⅎwnf 1811   ∈ wcel 2141  Ⅎwnfc 2908  ℩cio 6490  ℩crio 7366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-v 3455  df-ss 3921  df-sn 4589  df-uni 4872  df-iota 6492  df-riota 7367

This theorem is referenced by:  nfriota  7379