Theorem nfriotad 7127

Description: Deduction version of nfriota 7128. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker nfriotadw 7124 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 7116	. 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		nfriotad.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
3		nfnae 2456	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
4	2, 3	nfan 1900	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
5		nfcvf 3009	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
6	5	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦)
7		nfriotad.3	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝐴)
8	7	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴)
9	6, 8	nfeld 2991	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴)
10		nfriotad.2	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜓)
11	10	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
12	9, 11	nfand 1898	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
13	4, 12	nfiotad 6321	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
14	13	ex 415	. . 3 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))))
15		nfiota1 6318	. . . 4 ⊢ Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
16		eqidd 2824	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))
17	16	drnfc1 2999	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))))
18	15, 17	mpbiri 260	. . 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 2978	1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  ∀wal 1535  Ⅎwnf 1784   ∈ wcel 2114  Ⅎwnfc 2963  ℩cio 6314  ℩crio 7115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-v 3498  df-in 3945  df-ss 3954  df-sn 4570  df-uni 4841  df-iota 6316  df-riota 7116

This theorem is referenced by: (None)