Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ralprg | GIF version |
Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
ralprg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ralprg.2 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
ralprg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3529 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | 1 | raleqi 2628 | . . 3 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑) |
3 | ralunb 3252 | . . 3 ⊢ (∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑)) | |
4 | 2, 3 | bitri 183 | . 2 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑)) |
5 | ralprg.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 5 | ralsng 3559 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
7 | ralprg.2 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
8 | 7 | ralsng 3559 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (∀𝑥 ∈ {𝐵}𝜑 ↔ 𝜒)) |
9 | 6, 8 | bi2anan9 595 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∧ 𝜒))) |
10 | 4, 9 | syl5bb 191 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∀wral 2414 ∪ cun 3064 {csn 3522 {cpr 3523 |
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-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 df-sbc 2905 df-un 3070 df-sn 3528 df-pr 3529 |
This theorem is referenced by: raltpg 3571 ralpr 3573 iinxprg 3882 fvinim0ffz 10011 sumpr 11175 |
Copyright terms: Public domain | W3C validator |