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Theorem raldifb 3247
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 261 . . . 4 (((𝑥𝐴𝑥𝐵) → 𝜑) ↔ (𝑥𝐴 → (𝑥𝐵𝜑)))
21bicomi 131 . . 3 ((𝑥𝐴 → (𝑥𝐵𝜑)) ↔ ((𝑥𝐴𝑥𝐵) → 𝜑))
3 df-nel 2423 . . . . . 6 (𝑥𝐵 ↔ ¬ 𝑥𝐵)
43anbi2i 453 . . . . 5 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 eldif 3111 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
65bicomi 131 . . . . 5 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ 𝑥 ∈ (𝐴𝐵))
74, 6bitri 183 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐴𝐵))
87imbi1i 237 . . 3 (((𝑥𝐴𝑥𝐵) → 𝜑) ↔ (𝑥 ∈ (𝐴𝐵) → 𝜑))
92, 8bitri 183 . 2 ((𝑥𝐴 → (𝑥𝐵𝜑)) ↔ (𝑥 ∈ (𝐴𝐵) → 𝜑))
109ralbii2 2467 1 (∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥 ∈ (𝐴𝐵)𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wcel 2128  wnel 2422  wral 2435  cdif 3099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-nel 2423  df-ral 2440  df-v 2714  df-dif 3104
This theorem is referenced by: (None)
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