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Theorem dfrab3ss 3243
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 2959 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 ineq1 3159 . . . 4 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ∩ {𝑥𝜑}) = (𝐴 ∩ {𝑥𝜑}))
32eqcomd 2061 . . 3 ((𝐴𝐵) = 𝐴 → (𝐴 ∩ {𝑥𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑}))
41, 3sylbi 118 . 2 (𝐴𝐵 → (𝐴 ∩ {𝑥𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑}))
5 dfrab3 3241 . 2 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
6 dfrab3 3241 . . . 4 {𝑥𝐵𝜑} = (𝐵 ∩ {𝑥𝜑})
76ineq2i 3163 . . 3 (𝐴 ∩ {𝑥𝐵𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
8 inass 3175 . . 3 ((𝐴𝐵) ∩ {𝑥𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
97, 8eqtr4i 2079 . 2 (𝐴 ∩ {𝑥𝐵𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑})
104, 5, 93eqtr4g 2113 1 (𝐴𝐵 → {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝐵𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  {cab 2042  {crab 2327  cin 2944  wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576  df-in 2952  df-ss 2959
This theorem is referenced by: (None)
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