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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 3681 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)) |
3 | 2 | imbi1i 350 | . . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒)) |
4 | impexp 452 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) ↔ (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) | |
5 | 3, 4 | bitri 275 | . 2 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝜒) ↔ (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
6 | 5 | ralbii2 3090 | 1 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∀wral 3062 {crab 3433 |
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 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rab 3434 df-v 3477 |
This theorem is referenced by: frminex 5652 wereu2 5669 frpomin 6333 weniso 7338 zmin 12915 prmreclem1 16836 lublecllem 18300 mhmeql 18694 ghmeql 19100 pgpfac1lem5 19932 lmhmeql 20643 islindf4 21366 1stcfb 22918 fbssfi 23310 filssufilg 23384 txflf 23479 ptcmplem3 23527 symgtgp 23579 tgpconncompeqg 23585 cnllycmp 24441 ovolgelb 24966 dyadmax 25084 lhop1 25500 radcnvlt1 25899 noextenddif 27138 conway 27267 madebdaylemlrcut 27360 poimirlem4 36397 poimirlem32 36425 ismblfin 36434 igenval2 36840 glbconN 38153 glbconNOLD 38154 nadd2rabtr 42005 mgmhmeql 46446 intubeu 47449 unilbeu 47450 |
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