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Mirrors > Home > ILE Home > Th. List > ceqsex2 | GIF version |
Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) |
Ref | Expression |
---|---|
ceqsex2.1 | ⊢ Ⅎ𝑥𝜓 |
ceqsex2.2 | ⊢ Ⅎ𝑦𝜒 |
ceqsex2.3 | ⊢ 𝐴 ∈ V |
ceqsex2.4 | ⊢ 𝐵 ∈ V |
ceqsex2.5 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ceqsex2.6 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ceqsex2 | ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anass 966 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑))) | |
2 | 1 | exbii 1584 | . . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑))) |
3 | 19.42v 1878 | . . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑))) | |
4 | 2, 3 | bitri 183 | . . 3 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑))) |
5 | 4 | exbii 1584 | . 2 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑))) |
6 | nfv 1508 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝐵 | |
7 | ceqsex2.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
8 | 6, 7 | nfan 1544 | . . . 4 ⊢ Ⅎ𝑥(𝑦 = 𝐵 ∧ 𝜓) |
9 | 8 | nfex 1616 | . . 3 ⊢ Ⅎ𝑥∃𝑦(𝑦 = 𝐵 ∧ 𝜓) |
10 | ceqsex2.3 | . . 3 ⊢ 𝐴 ∈ V | |
11 | ceqsex2.5 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
12 | 11 | anbi2d 459 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑦 = 𝐵 ∧ 𝜓))) |
13 | 12 | exbidv 1797 | . . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦(𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝜓))) |
14 | 9, 10, 13 | ceqsex 2719 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝜓)) |
15 | ceqsex2.2 | . . 3 ⊢ Ⅎ𝑦𝜒 | |
16 | ceqsex2.4 | . . 3 ⊢ 𝐵 ∈ V | |
17 | ceqsex2.6 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
18 | 15, 16, 17 | ceqsex 2719 | . 2 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝜓) ↔ 𝜒) |
19 | 5, 14, 18 | 3bitri 205 | 1 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 Ⅎwnf 1436 ∃wex 1468 ∈ wcel 1480 Vcvv 2681 |
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-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-v 2683 |
This theorem is referenced by: ceqsex2v 2722 |
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