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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 5363) 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 3466 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | riotasv3d.7 | . . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝜓) | |
3 | 2 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ∃𝑦 ∈ 𝐵 𝜓) |
4 | riotasv3d.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
5 | nfv 1918 | . . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ V | |
6 | riotasv3d.5 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝜒)) | |
7 | 6 | imp 408 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝜓)) → 𝜒) |
8 | 7 | adantrl 715 | . . . . . . . 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 37447 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶)) |
12 | 11 | impr 456 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐷 = 𝐶) |
13 | 12 | eqcomd 2743 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐶 = 𝐷) |
14 | riotasv3d.4 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝜒 ↔ 𝜃)) | |
15 | 13, 14 | syldan 592 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → (𝜒 ↔ 𝜃)) |
16 | 8, 15 | mpbid 231 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝜃) |
17 | 16 | exp45 440 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ V → (𝑦 ∈ 𝐵 → (𝜓 → 𝜃)))) |
18 | 4, 5, 17 | ralrimd 3250 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ V → ∀𝑦 ∈ 𝐵 (𝜓 → 𝜃))) |
19 | riotasv3d.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑦𝜃) | |
20 | r19.23t 3241 | . . . . . 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 238 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ V → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃))) |
23 | 22 | imp 408 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃)) |
24 | 3, 23 | mpd 15 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝜃) |
25 | 1, 24 | sylan2 594 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 ∀wral 3065 ∃wrex 3074 Vcvv 3448 ℩crio 7317 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-riotaBAD 37444 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-riota 7318 df-undef 8209 |
This theorem is referenced by: cdlemefs32sn1aw 38906 cdleme43fsv1snlem 38912 cdleme41sn3a 38925 cdleme40m 38959 cdleme40n 38960 cdlemkid 39428 dihvalcqpre 39727 |
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