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Mirrors > Home > ILE Home > Th. List > 2ralunsn | GIF version |
Description: Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) |
Ref | Expression |
---|---|
2ralunsn.1 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) |
2ralunsn.2 | ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓)) |
2ralunsn.3 | ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜃)) |
Ref | Expression |
---|---|
2ralunsn | ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2ralunsn.2 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓)) | |
2 | 1 | ralunsn 3719 | . . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓))) |
3 | 2 | ralbidv 2435 | . 2 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓))) |
4 | 2ralunsn.1 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
5 | 4 | ralbidv 2435 | . . . . 5 ⊢ (𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜒)) |
6 | 2ralunsn.3 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜃)) | |
7 | 5, 6 | anbi12d 464 | . . . 4 ⊢ (𝑥 = 𝐵 → ((∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃))) |
8 | 7 | ralunsn 3719 | . . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃)))) |
9 | r19.26 2556 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | |
10 | 9 | anbi1i 453 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃)) ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃))) |
11 | 8, 10 | syl6bb 195 | . 2 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃)))) |
12 | 3, 11 | bitrd 187 | 1 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∀wral 2414 ∪ cun 3064 {csn 3522 |
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 |
This theorem is referenced by: (None) |
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