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Theorem dfrab3ss 3209
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 → {x Aφ} = (A ∩ {x Bφ}))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem dfrab3ss
StepHypRef Expression
1 df-ss 2925 . . 3 (AB ↔ (AB) = A)
2 ineq1 3125 . . . 4 ((AB) = A → ((AB) ∩ {xφ}) = (A ∩ {xφ}))
32eqcomd 2042 . . 3 ((AB) = A → (A ∩ {xφ}) = ((AB) ∩ {xφ}))
41, 3sylbi 114 . 2 (AB → (A ∩ {xφ}) = ((AB) ∩ {xφ}))
5 dfrab3 3207 . 2 {x Aφ} = (A ∩ {xφ})
6 dfrab3 3207 . . . 4 {x Bφ} = (B ∩ {xφ})
76ineq2i 3129 . . 3 (A ∩ {x Bφ}) = (A ∩ (B ∩ {xφ}))
8 inass 3141 . . 3 ((AB) ∩ {xφ}) = (A ∩ (B ∩ {xφ}))
97, 8eqtr4i 2060 . 2 (A ∩ {x Bφ}) = ((AB) ∩ {xφ})
104, 5, 93eqtr4g 2094 1 (AB → {x Aφ} = (A ∩ {x Bφ}))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  {cab 2023  {crab 2304  cin 2910  wss 2911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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