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Mirrors > Home > MPE Home > Th. List > raltp | Structured version Visualization version GIF version |
Description: Convert a quantification over a 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 4636 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))) |
8 | 1, 2, 3, 7 | mp3an 1457 | 1 ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 {ctp 4573 |
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 2795 |
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-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-v 3498 df-sbc 3775 df-un 3943 df-sn 4570 df-pr 4572 df-tp 4574 |
This theorem is referenced by: fztpval 12972 2wlkdlem4 27709 2pthdlem1 27711 3wlkdlem5 27944 3wlkdlem10 27950 upgr3v3e3cycl 27961 poimirlem9 34903 |
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