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Mirrors > Home > MPE Home > Th. List > nfriotad | Structured version Visualization version GIF version |
Description: Deduction version of nfriota 7140. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfriotadw 7135 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 7127 | . 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 1906 | . . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
5 | nfcvf 2928 | . . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
6 | 5 | adantl 485 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦) |
7 | nfriotad.3 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
8 | 7 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴) |
9 | 6, 8 | nfeld 2910 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴) |
10 | nfriotad.2 | . . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
11 | 10 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
12 | 9, 11 | nfand 1904 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
13 | 4, 12 | nfiotad 6302 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))) |
14 | 13 | ex 416 | . . 3 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)))) |
15 | nfiota1 6299 | . . . 4 ⊢ Ⅎ𝑦(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
16 | eqidd 2739 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))) | |
17 | 16 | drnfc1 2918 | . . . 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 184 | . 2 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))) |
20 | 1, 19 | nfcxfrd 2898 | 1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∀wal 1540 Ⅎwnf 1790 ∈ wcel 2114 Ⅎwnfc 2879 ℩cio 6295 ℩crio 7126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-13 2372 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3400 df-in 3850 df-ss 3860 df-sn 4517 df-uni 4797 df-iota 6297 df-riota 7127 |
This theorem is referenced by: (None) |
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