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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 3374 | . . . 4 ⊢ {A, B} = ({A} ∪ {B}) | |
2 | 1 | raleqi 2503 | . . 3 ⊢ (∀x ∈ {A, B}φ ↔ ∀x ∈ ({A} ∪ {B})φ) |
3 | ralunb 3118 | . . 3 ⊢ (∀x ∈ ({A} ∪ {B})φ ↔ (∀x ∈ {A}φ ∧ ∀x ∈ {B}φ)) | |
4 | 2, 3 | bitri 173 | . 2 ⊢ (∀x ∈ {A, B}φ ↔ (∀x ∈ {A}φ ∧ ∀x ∈ {B}φ)) |
5 | ralprg.1 | . . . 4 ⊢ (x = A → (φ ↔ ψ)) | |
6 | 5 | ralsng 3402 | . . 3 ⊢ (A ∈ 𝑉 → (∀x ∈ {A}φ ↔ ψ)) |
7 | ralprg.2 | . . . 4 ⊢ (x = B → (φ ↔ χ)) | |
8 | 7 | ralsng 3402 | . . 3 ⊢ (B ∈ 𝑊 → (∀x ∈ {B}φ ↔ χ)) |
9 | 6, 8 | bi2anan9 538 | . 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ((∀x ∈ {A}φ ∧ ∀x ∈ {B}φ) ↔ (ψ ∧ χ))) |
10 | 4, 9 | syl5bb 181 | 1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (∀x ∈ {A, B}φ ↔ (ψ ∧ χ))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ∀wral 2300 ∪ cun 2909 {csn 3367 {cpr 3368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-sbc 2759 df-un 2916 df-sn 3373 df-pr 3374 |
This theorem is referenced by: raltpg 3414 ralpr 3416 iinxprg 3722 |
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