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Theorem dfrab3ss 3533
 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 (A B → {x A φ} = (A ∩ {x B φ}))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem dfrab3ss
StepHypRef Expression
1 df-ss 3259 . . 3 (A B ↔ (AB) = A)
2 ineq1 3450 . . . 4 ((AB) = A → ((AB) ∩ {x φ}) = (A ∩ {x φ}))
32eqcomd 2358 . . 3 ((AB) = A → (A ∩ {x φ}) = ((AB) ∩ {x φ}))
41, 3sylbi 187 . 2 (A B → (A ∩ {x φ}) = ((AB) ∩ {x φ}))
5 dfrab3 3531 . 2 {x A φ} = (A ∩ {x φ})
6 dfrab3 3531 . . . 4 {x B φ} = (B ∩ {x φ})
76ineq2i 3454 . . 3 (A ∩ {x B φ}) = (A ∩ (B ∩ {x φ}))
8 inass 3465 . . 3 ((AB) ∩ {x φ}) = (A ∩ (B ∩ {x φ}))
97, 8eqtr4i 2376 . 2 (A ∩ {x B φ}) = ((AB) ∩ {x φ})
104, 5, 93eqtr4g 2410 1 (A B → {x A φ} = (A ∩ {x B φ}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  {cab 2339  {crab 2618   ∩ cin 3208   ⊆ wss 3257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by: (None)
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