![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rextp | GIF version |
Description: Convert a quantification over a 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 3645 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∨ 𝜒 ∨ 𝜃))) |
8 | 1, 2, 3, 7 | mp3an 1337 | 1 ⊢ (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∨ 𝜒 ∨ 𝜃)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 Vcvv 2737 {ctp 3593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-sn 3597 df-pr 3598 df-tp 3599 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |