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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 6474 | . . . 4 ⊢ (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅) | |
2 | ectocl.1 | . . . 4 ⊢ 𝑆 = (𝐵 / 𝑅) | |
3 | 1, 2 | eleq2s 2232 | . . 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 2140 | . . . . 5 ⊢ (𝐴 = [𝑥]𝑅 → (𝜑 ↔ 𝜓)) |
7 | 4, 6 | syl5ibcom 154 | . . . 4 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → (𝐴 = [𝑥]𝑅 → 𝜓)) |
8 | 7 | rexlimdva 2547 | . . 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 1331 ∈ wcel 1480 ∃wrex 2415 [cec 6420 / cqs 6421 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-qs 6428 |
This theorem is referenced by: ectocl 6489 elqsn0m 6490 qsel 6499 |
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