Theorem nfriotad 7378

Description: Deduction version of nfriota 7379. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker nfriotadw 7375 when possible. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
nfriotad	⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))

Proof of Theorem nfriotad

Step	Hyp	Ref	Expression
1		df-riota 7367	. 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		nfriotad.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
3		nfnae 2466	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
4	2, 3	nfan 1929	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
5		nfcvf 2951	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
6	5	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦)
7		nfriotad.3	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝐴)
8	7	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴)
9	6, 8	nfeld 2936	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴)
10		nfriotad.2	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜓)
11	10	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
12	9, 11	nfand 1927	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
13	4, 12	nfiotad 6497	. . . 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		eqidd 2764	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
17	16	drnfc1 2944	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))))
18	15, 17	mpbiri 261	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
19	14, 18	pm2.61d2 183	. 2 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
20	1, 19	nfcxfrd 2924	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1568  Ⅎwnf 1813   ∈ wcel 2143  Ⅎwnfc 2910  ℩cio 6490  ℩crio 7366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-v 3457  df-ss 3922  df-sn 4590  df-uni 4873  df-iota 6492  df-riota 7367

This theorem is used by: (None)