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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 1368 | . 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 277 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝜑) |
6 | 2, 3, 5 | ectocld 6655 | . 2 ⊢ ((⊤ ∧ 𝐴 ∈ 𝑆) → 𝜓) |
7 | 1, 6 | mpan 424 | 1 ⊢ (𝐴 ∈ 𝑆 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ⊤wtru 1365 ∈ wcel 2164 [cec 6585 / cqs 6586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-qs 6593 |
This theorem is referenced by: (None) |
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