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Theorem dfrab3ss 3301
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 3034 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 ineq1 3217 . . . 4 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ∩ {𝑥𝜑}) = (𝐴 ∩ {𝑥𝜑}))
32eqcomd 2105 . . 3 ((𝐴𝐵) = 𝐴 → (𝐴 ∩ {𝑥𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑}))
41, 3sylbi 120 . 2 (𝐴𝐵 → (𝐴 ∩ {𝑥𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑}))
5 dfrab3 3299 . 2 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
6 dfrab3 3299 . . . 4 {𝑥𝐵𝜑} = (𝐵 ∩ {𝑥𝜑})
76ineq2i 3221 . . 3 (𝐴 ∩ {𝑥𝐵𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
8 inass 3233 . . 3 ((𝐴𝐵) ∩ {𝑥𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
97, 8eqtr4i 2123 . 2 (𝐴 ∩ {𝑥𝐵𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑})
104, 5, 93eqtr4g 2157 1 (𝐴𝐵 → {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝐵𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  {cab 2086  {crab 2379  cin 3020  wss 3021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rab 2384  df-v 2643  df-in 3027  df-ss 3034
This theorem is referenced by: (None)
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