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Mirrors > Home > MPE Home > Th. List > rextp | Structured version Visualization version GIF version |
Description: Convert an existential quantification over an unordered triple to a disjunction. (Contributed 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 |
---|---|
rextp | ⊢ (∃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 | rextpg 4637 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∨ 𝜒 ∨ 𝜃))) |
8 | 1, 2, 3, 7 | mp3an 1460 | 1 ⊢ (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∨ 𝜒 ∨ 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ w3o 1085 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 Vcvv 3431 {ctp 4567 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-v 3433 df-un 3893 df-sn 4564 df-pr 4566 df-tp 4568 |
This theorem is referenced by: 1cubr 25981 |
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