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Mirrors > Home > ILE Home > Th. List > ectocl | GIF version |
Description: Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
ectocl.1 | ⊢ 𝑆 = (𝐵 / 𝑅) |
ectocl.2 | ⊢ ([𝑥]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) |
ectocl.3 | ⊢ (𝑥 ∈ 𝐵 → 𝜑) |
Ref | Expression |
---|---|
ectocl | ⊢ (𝐴 ∈ 𝑆 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1288 | . 2 ⊢ ⊤ | |
2 | ectocl.1 | . . 3 ⊢ 𝑆 = (𝐵 / 𝑅) | |
3 | ectocl.2 | . . 3 ⊢ ([𝑥]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | ectocl.3 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝜑) | |
5 | 4 | adantl 271 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝜑) |
6 | 2, 3, 5 | ectocld 6195 | . 2 ⊢ ((⊤ ∧ 𝐴 ∈ 𝑆) → 𝜓) |
7 | 1, 6 | mpan 414 | 1 ⊢ (𝐴 ∈ 𝑆 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ⊤wtru 1285 ∈ wcel 1433 [cec 6127 / cqs 6128 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-qs 6135 |
This theorem is referenced by: (None) |
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