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| Mirrors > Home > MPE Home > Th. List > raltp | Structured version Visualization version GIF version | ||
| Description: Convert a universal quantification over an unordered triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| raltp.1 | ⊢ 𝐴 ∈ V | 
| raltp.2 | ⊢ 𝐵 ∈ V | 
| raltp.3 | ⊢ 𝐶 ∈ V | 
| raltp.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| raltp.5 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | 
| raltp.6 | ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃)) | 
| Ref | Expression | 
|---|---|
| raltp | ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | raltp.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | raltp.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | raltp.3 | . 2 ⊢ 𝐶 ∈ V | |
| 4 | raltp.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 5 | raltp.5 | . . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
| 6 | raltp.6 | . . 3 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃)) | |
| 7 | 4, 5, 6 | raltpg 4698 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))) | 
| 8 | 1, 2, 3, 7 | mp3an 1463 | 1 ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 {ctp 4630 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 df-tp 4631 | 
| This theorem is referenced by: fztpval 13626 2wlkdlem4 29948 2pthdlem1 29950 3wlkdlem5 30182 3wlkdlem10 30188 upgr3v3e3cycl 30199 poimirlem9 37636 cycl3grtrilem 47913 usgrexmpl2lem 47985 usgrexmpl2trifr 47996 gpg5nbgrvtx03star 48036 gpg5nbgr3star 48037 | 
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