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Theorem dfrab3ss 3425
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 3154 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 ineq1 3341 . . . 4 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ∩ {𝑥𝜑}) = (𝐴 ∩ {𝑥𝜑}))
32eqcomd 2193 . . 3 ((𝐴𝐵) = 𝐴 → (𝐴 ∩ {𝑥𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑}))
41, 3sylbi 121 . 2 (𝐴𝐵 → (𝐴 ∩ {𝑥𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑}))
5 dfrab3 3423 . 2 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
6 dfrab3 3423 . . . 4 {𝑥𝐵𝜑} = (𝐵 ∩ {𝑥𝜑})
76ineq2i 3345 . . 3 (𝐴 ∩ {𝑥𝐵𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
8 inass 3357 . . 3 ((𝐴𝐵) ∩ {𝑥𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
97, 8eqtr4i 2211 . 2 (𝐴 ∩ {𝑥𝐵𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑})
104, 5, 93eqtr4g 2245 1 (𝐴𝐵 → {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝐵𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  {cab 2173  {crab 2469  cin 3140  wss 3141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-v 2751  df-in 3147  df-ss 3154
This theorem is referenced by: (None)
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