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Theorem dfrab3ss 4233
 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 3898 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 ineq1 4131 . . . 4 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ∩ {𝑥𝜑}) = (𝐴 ∩ {𝑥𝜑}))
32eqcomd 2804 . . 3 ((𝐴𝐵) = 𝐴 → (𝐴 ∩ {𝑥𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑}))
41, 3sylbi 220 . 2 (𝐴𝐵 → (𝐴 ∩ {𝑥𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑}))
5 dfrab3 4230 . 2 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
6 dfrab3 4230 . . . 4 {𝑥𝐵𝜑} = (𝐵 ∩ {𝑥𝜑})
76ineq2i 4136 . . 3 (𝐴 ∩ {𝑥𝐵𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
8 inass 4146 . . 3 ((𝐴𝐵) ∩ {𝑥𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
97, 8eqtr4i 2824 . 2 (𝐴 ∩ {𝑥𝐵𝜑}) = ((𝐴𝐵) ∩ {𝑥𝜑})
104, 5, 93eqtr4g 2858 1 (𝐴𝐵 → {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝐵𝜑}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  {cab 2776  {crab 3110   ∩ cin 3880   ⊆ wss 3881 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898 This theorem is referenced by:  mbfposadd  35123  proot1hash  40187
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