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Mirrors > Home > ILE Home > Th. List > rextpg | GIF version |
Description: Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
ralprg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ralprg.2 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) |
raltpg.3 | ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃)) |
Ref | Expression |
---|---|
rextpg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∨ 𝜒 ∨ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralprg.1 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | ralprg.2 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
3 | 1, 2 | rexprg 3570 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒))) |
4 | 3 | orbi1d 780 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓 ∨ 𝜒) ∨ ∃𝑥 ∈ {𝐶}𝜑))) |
5 | raltpg.3 | . . . . . 6 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃)) | |
6 | 5 | rexsng 3560 | . . . . 5 ⊢ (𝐶 ∈ 𝑋 → (∃𝑥 ∈ {𝐶}𝜑 ↔ 𝜃)) |
7 | 6 | orbi2d 779 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → (((𝜓 ∨ 𝜒) ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓 ∨ 𝜒) ∨ 𝜃))) |
8 | 4, 7 | sylan9bb 457 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓 ∨ 𝜒) ∨ 𝜃))) |
9 | 8 | 3impa 1176 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓 ∨ 𝜒) ∨ 𝜃))) |
10 | df-tp 3530 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
11 | 10 | rexeqi 2629 | . . 3 ⊢ (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ ∃𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑) |
12 | rexun 3251 | . . 3 ⊢ (∃𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑)) | |
13 | 11, 12 | bitri 183 | . 2 ⊢ (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑)) |
14 | df-3or 963 | . 2 ⊢ ((𝜓 ∨ 𝜒 ∨ 𝜃) ↔ ((𝜓 ∨ 𝜒) ∨ 𝜃)) | |
15 | 9, 13, 14 | 3bitr4g 222 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∨ 𝜒 ∨ 𝜃))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 697 ∨ w3o 961 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ∃wrex 2415 ∪ cun 3064 {csn 3522 {cpr 3523 {ctp 3524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-sn 3528 df-pr 3529 df-tp 3530 |
This theorem is referenced by: rextp 3576 |
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