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| Description: Deduction version of nfriota 7401. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker nfriotadw 7397 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 7389 | . 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
| 2 | nfriotad.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfnae 2438 | . . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
| 4 | 2, 3 | nfan 1898 | . . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) | 
| 5 | nfcvf 2931 | . . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
| 6 | 5 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦) | 
| 7 | nfriotad.3 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 8 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴) | 
| 9 | 6, 8 | nfeld 2916 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴) | 
| 10 | nfriotad.2 | . . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 11 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | 
| 12 | 9, 11 | nfand 1896 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) | 
| 13 | 4, 12 | nfiotad 6518 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))) | 
| 14 | 13 | ex 412 | . . 3 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))) | 
| 15 | nfiota1 6515 | . . . 4 ⊢ Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
| 16 | eqidd 2737 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))) | |
| 17 | 16 | drnfc1 2924 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))) | 
| 18 | 15, 17 | mpbiri 258 | . . 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 2903 | 1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1537 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2889 ℩cio 6511 ℩crio 7388 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-13 2376 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-v 3481 df-ss 3967 df-sn 4626 df-uni 4907 df-iota 6513 df-riota 7389 | 
| This theorem is referenced by: (None) | 
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