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Mirrors > Home > MPE Home > Th. List > ralrab | Structured version Visualization version GIF version |
Description: Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
ralab.1 | ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ralrab | ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralab.1 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) | |
2 | 1 | elrab 3683 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)) |
3 | 2 | imbi1i 349 | . . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒)) |
4 | impexp 451 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) ↔ (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) | |
5 | 3, 4 | bitri 274 | . 2 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝜒) ↔ (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
6 | 5 | ralbii2 3089 | 1 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ∀wral 3061 {crab 3432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rab 3433 df-v 3476 |
This theorem is referenced by: frminex 5656 wereu2 5673 frpomin 6341 weniso 7353 zmin 12932 prmreclem1 16853 lublecllem 18317 mgmhmeql 18641 mhmeql 18743 ghmeql 19153 pgpfac1lem5 19990 lmhmeql 20810 islindf4 21612 1stcfb 23169 fbssfi 23561 filssufilg 23635 txflf 23730 ptcmplem3 23778 symgtgp 23830 tgpconncompeqg 23836 cnllycmp 24696 ovolgelb 25221 dyadmax 25339 lhop1 25755 radcnvlt1 26154 noextenddif 27395 conway 27525 madebdaylemlrcut 27618 poimirlem4 36795 poimirlem32 36823 ismblfin 36832 igenval2 37237 glbconN 38550 glbconNOLD 38551 nadd2rabtr 42436 intubeu 47697 unilbeu 47698 |
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