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Mirrors > Home > ILE Home > Th. List > eqvincf | GIF version |
Description: A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.) |
Ref | Expression |
---|---|
eqvincf.1 | ⊢ Ⅎ𝑥𝐴 |
eqvincf.2 | ⊢ Ⅎ𝑥𝐵 |
eqvincf.3 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
eqvincf | ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvincf.3 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | eqvinc 2808 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵)) |
3 | eqvincf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | nfeq2 2293 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
5 | eqvincf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | nfeq2 2293 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵 |
7 | 4, 6 | nfan 1544 | . . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 = 𝐵) |
8 | nfv 1508 | . . 3 ⊢ Ⅎ𝑦(𝑥 = 𝐴 ∧ 𝑥 = 𝐵) | |
9 | eqeq1 2146 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴)) | |
10 | eqeq1 2146 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐵 ↔ 𝑥 = 𝐵)) | |
11 | 9, 10 | anbi12d 464 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝐵))) |
12 | 7, 8, 11 | cbvex 1729 | . 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
13 | 2, 12 | bitri 183 | 1 ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Ⅎwnfc 2268 Vcvv 2686 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 |
This theorem is referenced by: (None) |
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