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Theorem inrab 3527
 Description: Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
Assertion
Ref Expression
inrab ({x A φ} ∩ {x A ψ}) = {x A (φ ψ)}

Proof of Theorem inrab
StepHypRef Expression
1 df-rab 2623 . . 3 {x A φ} = {x (x A φ)}
2 df-rab 2623 . . 3 {x A ψ} = {x (x A ψ)}
31, 2ineq12i 3455 . 2 ({x A φ} ∩ {x A ψ}) = ({x (x A φ)} ∩ {x (x A ψ)})
4 df-rab 2623 . . 3 {x A (φ ψ)} = {x (x A (φ ψ))}
5 inab 3522 . . . 4 ({x (x A φ)} ∩ {x (x A ψ)}) = {x ((x A φ) (x A ψ))}
6 anandi 801 . . . . 5 ((x A (φ ψ)) ↔ ((x A φ) (x A ψ)))
76abbii 2465 . . . 4 {x (x A (φ ψ))} = {x ((x A φ) (x A ψ))}
85, 7eqtr4i 2376 . . 3 ({x (x A φ)} ∩ {x (x A ψ)}) = {x (x A (φ ψ))}
94, 8eqtr4i 2376 . 2 {x A (φ ψ)} = ({x (x A φ)} ∩ {x (x A ψ)})
103, 9eqtr4i 2376 1 ({x A φ} ∩ {x A ψ}) = {x A (φ ψ)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618   ∩ cin 3208 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 This theorem is referenced by:  rabnc  3574
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