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Theorem raldifb 3124
Description: Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
Assertion
Ref Expression
raldifb (∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥 ∈ (𝐴𝐵)𝜑)

Proof of Theorem raldifb
StepHypRef Expression
1 impexp 259 . . . 4 (((𝑥𝐴𝑥𝐵) → 𝜑) ↔ (𝑥𝐴 → (𝑥𝐵𝜑)))
21bicomi 130 . . 3 ((𝑥𝐴 → (𝑥𝐵𝜑)) ↔ ((𝑥𝐴𝑥𝐵) → 𝜑))
3 df-nel 2345 . . . . . 6 (𝑥𝐵 ↔ ¬ 𝑥𝐵)
43anbi2i 445 . . . . 5 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 eldif 2993 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
65bicomi 130 . . . . 5 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ 𝑥 ∈ (𝐴𝐵))
74, 6bitri 182 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐴𝐵))
87imbi1i 236 . . 3 (((𝑥𝐴𝑥𝐵) → 𝜑) ↔ (𝑥 ∈ (𝐴𝐵) → 𝜑))
92, 8bitri 182 . 2 ((𝑥𝐴 → (𝑥𝐵𝜑)) ↔ (𝑥 ∈ (𝐴𝐵) → 𝜑))
109ralbii2 2382 1 (∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥 ∈ (𝐴𝐵)𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wcel 1434  wnel 2344  wral 2353  cdif 2981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-nel 2345  df-ral 2358  df-v 2614  df-dif 2986
This theorem is referenced by: (None)
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