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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 3795 | . . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓))) |
3 | 2 | ralbidv 2477 | . 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 2477 | . . . . 5 ⊢ (𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜒)) |
6 | 2ralunsn.3 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜃)) | |
7 | 5, 6 | anbi12d 473 | . . . 4 ⊢ (𝑥 = 𝐵 → ((∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃))) |
8 | 7 | ralunsn 3795 | . . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃)))) |
9 | r19.26 2603 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | |
10 | 9 | anbi1i 458 | . . 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 | bitrdi 196 | . 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 188 | 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 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∪ cun 3127 {csn 3591 |
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-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-sbc 2963 df-un 3133 df-sn 3597 |
This theorem is referenced by: (None) |
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