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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 6489 | . . . 4 ⊢ (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅) | |
2 | ectocl.1 | . . . 4 ⊢ 𝑆 = (𝐵 / 𝑅) | |
3 | 1, 2 | eleq2s 2235 | . . 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 2143 | . . . . 5 ⊢ (𝐴 = [𝑥]𝑅 → (𝜑 ↔ 𝜓)) |
7 | 4, 6 | syl5ibcom 154 | . . . 4 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → (𝐴 = [𝑥]𝑅 → 𝜓)) |
8 | 7 | rexlimdva 2552 | . . 3 ⊢ (𝜒 → (∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅 → 𝜓)) |
9 | 3, 8 | syl5 32 | . 2 ⊢ (𝜒 → (𝐴 ∈ 𝑆 → 𝜓)) |
10 | 9 | imp 123 | 1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 ∃wrex 2418 [cec 6435 / cqs 6436 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-qs 6443 |
This theorem is referenced by: ectocl 6504 elqsn0m 6505 qsel 6514 |
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