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Mirrors > Home > MPE Home > Th. List > nfriotad | Structured version Visualization version GIF version |
Description: Deduction version of nfriota 7245. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfriotadw 7240 when possible. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfriotad.1 | ⊢ Ⅎ𝑦𝜑 |
nfriotad.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
nfriotad.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfriotad | ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-riota 7232 | . 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
2 | nfriotad.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
3 | nfnae 2434 | . . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
4 | 2, 3 | nfan 1902 | . . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
5 | nfcvf 2936 | . . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
6 | 5 | adantl 482 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦) |
7 | nfriotad.3 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
8 | 7 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴) |
9 | 6, 8 | nfeld 2918 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴) |
10 | nfriotad.2 | . . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
11 | 10 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
12 | 9, 11 | nfand 1900 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
13 | 4, 12 | nfiotad 6396 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))) |
14 | 13 | ex 413 | . . 3 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))) |
15 | nfiota1 6393 | . . . 4 ⊢ Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
16 | eqidd 2739 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))) | |
17 | 16 | drnfc1 2926 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))) |
18 | 15, 17 | mpbiri 257 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))) |
19 | 14, 18 | pm2.61d2 181 | . 2 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))) |
20 | 1, 19 | nfcxfrd 2906 | 1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∀wal 1537 Ⅎwnf 1786 ∈ wcel 2106 Ⅎwnfc 2887 ℩cio 6389 ℩crio 7231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2372 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-v 3434 df-in 3894 df-ss 3904 df-sn 4562 df-uni 4840 df-iota 6391 df-riota 7232 |
This theorem is referenced by: (None) |
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