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Mirrors > Home > MPE Home > Th. List > ectocl | Structured version Visualization version 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 1537 | . 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 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝜑) |
6 | 2, 3, 5 | ectocld 8775 | . 2 ⊢ ((⊤ ∧ 𝐴 ∈ 𝑆) → 𝜓) |
7 | 1, 6 | mpan 687 | 1 ⊢ (𝐴 ∈ 𝑆 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 [cec 8698 / cqs 8699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rex 3063 df-qs 8706 |
This theorem is referenced by: vitalilem2 25482 |
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