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Mirrors > Home > MPE Home > Th. List > Mathboxes > riotasv3d | Structured version Visualization version GIF version |
Description: A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 4800) also holds for its description binder 𝐷 (in the form of property 𝜃). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
riotasv3d.1 | ⊢ Ⅎ𝑦𝜑 |
riotasv3d.2 | ⊢ (𝜑 → Ⅎ𝑦𝜃) |
riotasv3d.3 | ⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))) |
riotasv3d.4 | ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝜒 ↔ 𝜃)) |
riotasv3d.5 | ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝜒)) |
riotasv3d.6 | ⊢ (𝜑 → 𝐷 ∈ 𝐴) |
riotasv3d.7 | ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝜓) |
Ref | Expression |
---|---|
riotasv3d | ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | riotasv3d.7 | . . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝜓) | |
3 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ∃𝑦 ∈ 𝐵 𝜓) |
4 | riotasv3d.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
5 | nfv 1830 | . . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ V | |
6 | riotasv3d.5 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝜒)) | |
7 | 6 | imp 444 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝜓)) → 𝜒) |
8 | 7 | adantrl 748 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝜒) |
9 | riotasv3d.3 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))) | |
10 | riotasv3d.6 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ 𝐴) | |
11 | 9, 10 | riotasvd 33260 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶)) |
12 | 11 | impr 647 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐷 = 𝐶) |
13 | 12 | eqcomd 2616 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐶 = 𝐷) |
14 | riotasv3d.4 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝜒 ↔ 𝜃)) | |
15 | 13, 14 | syldan 486 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → (𝜒 ↔ 𝜃)) |
16 | 8, 15 | mpbid 221 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝜃) |
17 | 16 | exp45 640 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ V → (𝑦 ∈ 𝐵 → (𝜓 → 𝜃)))) |
18 | 4, 5, 17 | ralrimd 2942 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ V → ∀𝑦 ∈ 𝐵 (𝜓 → 𝜃))) |
19 | riotasv3d.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑦𝜃) | |
20 | r19.23t 3003 | . . . . . 6 ⊢ (Ⅎ𝑦𝜃 → (∀𝑦 ∈ 𝐵 (𝜓 → 𝜃) ↔ (∃𝑦 ∈ 𝐵 𝜓 → 𝜃))) | |
21 | 19, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝜓 → 𝜃) ↔ (∃𝑦 ∈ 𝐵 𝜓 → 𝜃))) |
22 | 18, 21 | sylibd 228 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ V → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃))) |
23 | 22 | imp 444 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃)) |
24 | 3, 23 | mpd 15 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝜃) |
25 | 1, 24 | sylan2 490 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ℩crio 6510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-riota 6511 df-undef 7286 |
This theorem is referenced by: cdlemefs32sn1aw 34720 cdleme43fsv1snlem 34726 cdleme41sn3a 34739 cdleme40m 34773 cdleme40n 34774 cdlemkid 35242 dihvalcqpre 35542 |
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