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Mirrors > Home > MPE Home > Th. List > rspc3ev | Structured version Visualization version GIF version |
Description: 3-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.) |
Ref | Expression |
---|---|
rspc3v.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) |
rspc3v.2 | ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) |
rspc3v.3 | ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓)) |
Ref | Expression |
---|---|
rspc3ev | ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑇 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1064 | . 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → 𝐴 ∈ 𝑅) | |
2 | simpl2 1065 | . 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → 𝐵 ∈ 𝑆) | |
3 | rspc3v.3 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓)) | |
4 | 3 | rspcev 3309 | . . 3 ⊢ ((𝐶 ∈ 𝑇 ∧ 𝜓) → ∃𝑧 ∈ 𝑇 𝜃) |
5 | 4 | 3ad2antl3 1225 | . 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → ∃𝑧 ∈ 𝑇 𝜃) |
6 | rspc3v.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
7 | 6 | rexbidv 3052 | . . 3 ⊢ (𝑥 = 𝐴 → (∃𝑧 ∈ 𝑇 𝜑 ↔ ∃𝑧 ∈ 𝑇 𝜒)) |
8 | rspc3v.2 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) | |
9 | 8 | rexbidv 3052 | . . 3 ⊢ (𝑦 = 𝐵 → (∃𝑧 ∈ 𝑇 𝜒 ↔ ∃𝑧 ∈ 𝑇 𝜃)) |
10 | 7, 9 | rspc2ev 3324 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∃𝑧 ∈ 𝑇 𝜃) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑇 𝜑) |
11 | 1, 2, 5, 10 | syl3anc 1326 | 1 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑇 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-v 3202 |
This theorem is referenced by: f1dom3el3dif 6526 wrdl3s3 13705 pmltpclem1 23217 axlowdim 25841 axeuclidlem 25842 upgr3v3e3cycl 27040 br8d 29422 tgoldbachgt 30741 br8 31646 br6 31647 3dim1lem5 34752 lplni2 34823 jm2.27 37575 |
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