![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 1546 | . 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 483 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝜑) |
6 | 2, 3, 5 | ectocld 8724 | . 2 ⊢ ((⊤ ∧ 𝐴 ∈ 𝑆) → 𝜓) |
7 | 1, 6 | mpan 689 | 1 ⊢ (𝐴 ∈ 𝑆 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 [cec 8647 / cqs 8648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rex 3075 df-qs 8655 |
This theorem is referenced by: vitalilem2 24976 |
Copyright terms: Public domain | W3C validator |