Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3331 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)) |
3 | 2 | imbi1i 338 | . . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒)) |
4 | impexp 461 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) ↔ (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) | |
5 | 3, 4 | bitri 263 | . 2 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝜒) ↔ (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
6 | 5 | ralbii2 2961 | 1 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 {crab 2900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 |
This theorem is referenced by: frminex 5018 wereu2 5035 weniso 6504 zmin 11660 prmreclem1 15458 lublecllem 16811 mhmeql 17187 ghmeql 17506 pgpfac1lem5 18301 lmhmeql 18876 islindf4 19996 1stcfb 21058 fbssfi 21451 filssufilg 21525 txflf 21620 ptcmplem3 21668 symgtgp 21715 tgpconcompeqg 21725 cnllycmp 22563 ovolgelb 23055 dyadmax 23172 lhop1 23581 radcnvlt1 23976 uvtxnb 26025 2spotdisj 26588 poimirlem4 32583 poimirlem32 32611 ismblfin 32620 igenval2 33035 glbconN 33681 mgmhmeql 41593 |
Copyright terms: Public domain | W3C validator |