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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 2667 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵)) |
3 | eqvincf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | nfeq2 2189 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
5 | eqvincf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | nfeq2 2189 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵 |
7 | 4, 6 | nfan 1457 | . . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 = 𝐵) |
8 | nfv 1421 | . . 3 ⊢ Ⅎ𝑦(𝑥 = 𝐴 ∧ 𝑥 = 𝐵) | |
9 | eqeq1 2046 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴)) | |
10 | eqeq1 2046 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐵 ↔ 𝑥 = 𝐵)) | |
11 | 9, 10 | anbi12d 442 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝐵))) |
12 | 7, 8, 11 | cbvex 1639 | . 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
13 | 2, 12 | bitri 173 | 1 ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ∃wex 1381 ∈ wcel 1393 Ⅎwnfc 2165 Vcvv 2557 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 |
This theorem is referenced by: (None) |
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