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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 5304) 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 3512 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | riotasv3d.7 | . . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝜓) | |
3 | 2 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ∃𝑦 ∈ 𝐵 𝜓) |
4 | riotasv3d.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
5 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ V | |
6 | riotasv3d.5 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝜒)) | |
7 | 6 | imp 409 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝜓)) → 𝜒) |
8 | 7 | adantrl 714 | . . . . . . . 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 36107 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶)) |
12 | 11 | impr 457 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐷 = 𝐶) |
13 | 12 | eqcomd 2827 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐶 = 𝐷) |
14 | riotasv3d.4 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝜒 ↔ 𝜃)) | |
15 | 13, 14 | syldan 593 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → (𝜒 ↔ 𝜃)) |
16 | 8, 15 | mpbid 234 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝜃) |
17 | 16 | exp45 441 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ V → (𝑦 ∈ 𝐵 → (𝜓 → 𝜃)))) |
18 | 4, 5, 17 | ralrimd 3218 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ V → ∀𝑦 ∈ 𝐵 (𝜓 → 𝜃))) |
19 | riotasv3d.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑦𝜃) | |
20 | r19.23t 3313 | . . . . . 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 241 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ V → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃))) |
23 | 22 | imp 409 | . . 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 208 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ℩crio 7113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-riota 7114 df-undef 7939 |
This theorem is referenced by: cdlemefs32sn1aw 37565 cdleme43fsv1snlem 37571 cdleme41sn3a 37584 cdleme40m 37618 cdleme40n 37619 cdlemkid 38087 dihvalcqpre 38386 |
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