Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ectocld | Structured version Visualization version 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 | ectocld.3 | . . . 4 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → 𝜑) | |
2 | ectocl.2 | . . . . 5 ⊢ ([𝑥]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | eqcoms 2744 | . . . 4 ⊢ (𝐴 = [𝑥]𝑅 → (𝜑 ↔ 𝜓)) |
4 | 1, 3 | syl5ibcom 248 | . . 3 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → (𝐴 = [𝑥]𝑅 → 𝜓)) |
5 | 4 | rexlimdva 3193 | . 2 ⊢ (𝜒 → (∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅 → 𝜓)) |
6 | elqsi 8430 | . . 3 ⊢ (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅) | |
7 | ectocl.1 | . . 3 ⊢ 𝑆 = (𝐵 / 𝑅) | |
8 | 6, 7 | eleq2s 2849 | . 2 ⊢ (𝐴 ∈ 𝑆 → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅) |
9 | 5, 8 | impel 509 | 1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 [cec 8367 / cqs 8368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-qs 8375 |
This theorem is referenced by: ectocl 8445 elqsn0 8446 qsdisj 8454 qsel 8456 eqgen 18551 orbsta 18661 sylow1lem3 18943 sylow2alem2 18961 sylow2a 18962 sylow2blem2 18964 frgpup1 19119 frgpup3lem 19121 quscrng 20232 pi1xfr 23906 pi1coghm 23912 vitalilem3 24461 qsdisjALTV 36414 eqvrelqsel 36415 |
Copyright terms: Public domain | W3C validator |