ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfrab3ss GIF version

Theorem dfrab3ss 3359
Description: Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)
Assertion
Ref Expression
dfrab3ss (𝐴𝐵 → {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝐵𝜑}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfrab3ss
StepHypRef Expression
1 df-ss 3089 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 ineq1 3275 . . . 4 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ∩ {𝑥𝜑}) = (𝐴 ∩ {𝑥𝜑}))
32eqcomd 2146 . . 3 ((𝐴𝐵) = 𝐴 → (𝐴 ∩ {𝑥𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑}))
41, 3sylbi 120 . 2 (𝐴𝐵 → (𝐴 ∩ {𝑥𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑}))
5 dfrab3 3357 . 2 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
6 dfrab3 3357 . . . 4 {𝑥𝐵𝜑} = (𝐵 ∩ {𝑥𝜑})
76ineq2i 3279 . . 3 (𝐴 ∩ {𝑥𝐵𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
8 inass 3291 . . 3 ((𝐴𝐵) ∩ {𝑥𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
97, 8eqtr4i 2164 . 2 (𝐴 ∩ {𝑥𝐵𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑})
104, 5, 93eqtr4g 2198 1 (𝐴𝐵 → {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝐵𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  {cab 2126  {crab 2421  cin 3075  wss 3076
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691  df-in 3082  df-ss 3089
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator