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Mirrors > Home > ILE Home > Th. List > ectocld | GIF version |
Description: Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
ectocl.1 | ⊢ 𝑆 = (𝐵 / 𝑅) |
ectocl.2 | ⊢ ([𝑥]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) |
ectocld.3 | ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → 𝜑) |
Ref | Expression |
---|---|
ectocld | ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqsi 6158 | . . . 4 ⊢ (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅) | |
2 | ectocl.1 | . . . 4 ⊢ 𝑆 = (𝐵 / 𝑅) | |
3 | 1, 2 | eleq2s 2132 | . . 3 ⊢ (𝐴 ∈ 𝑆 → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅) |
4 | ectocld.3 | . . . . 5 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → 𝜑) | |
5 | ectocl.2 | . . . . . 6 ⊢ ([𝑥]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 5 | eqcoms 2043 | . . . . 5 ⊢ (𝐴 = [𝑥]𝑅 → (𝜑 ↔ 𝜓)) |
7 | 4, 6 | syl5ibcom 144 | . . . 4 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → (𝐴 = [𝑥]𝑅 → 𝜓)) |
8 | 7 | rexlimdva 2433 | . . 3 ⊢ (𝜒 → (∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅 → 𝜓)) |
9 | 3, 8 | syl5 28 | . 2 ⊢ (𝜒 → (𝐴 ∈ 𝑆 → 𝜓)) |
10 | 9 | imp 115 | 1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 ∃wrex 2307 [cec 6104 / cqs 6105 |
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-ral 2311 df-rex 2312 df-v 2559 df-qs 6112 |
This theorem is referenced by: ectocl 6173 elqsn0m 6174 qsel 6183 |
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