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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 2734 | . . . 4 ⊢ (𝐴 = [𝑥]𝑅 → (𝜑 ↔ 𝜓)) |
4 | 1, 3 | syl5ibcom 244 | . . 3 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → (𝐴 = [𝑥]𝑅 → 𝜓)) |
5 | 4 | rexlimdva 3145 | . 2 ⊢ (𝜒 → (∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅 → 𝜓)) |
6 | elqsi 8801 | . . 3 ⊢ (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅) | |
7 | ectocl.1 | . . 3 ⊢ 𝑆 = (𝐵 / 𝑅) | |
8 | 6, 7 | eleq2s 2844 | . 2 ⊢ (𝐴 ∈ 𝑆 → ∃𝑥 ∈ 𝐵 𝐴 = [𝑥]𝑅) |
9 | 5, 8 | impel 504 | 1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∃wrex 3060 [cec 8734 / cqs 8735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rex 3061 df-qs 8742 |
This theorem is referenced by: ectocl 8816 elqsn0 8817 qsdisj 8825 qsel 8827 eqgen 19177 orbsta 19309 sylow1lem3 19600 sylow2alem2 19618 sylow2a 19619 sylow2blem2 19621 frgpup1 19775 frgpup3lem 19777 quscrng 21274 pi1xfr 25076 pi1coghm 25082 vitalilem3 25633 qsdisjALTV 38315 eqvrelqsel 38316 |
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