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Theorem iinab 4027
 Description: Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
Assertion
Ref Expression
iinab x A {y φ} = {y x A φ}
Distinct variable groups:   y,A   x,y
Allowed substitution hints:   φ(x,y)   A(x)

Proof of Theorem iinab
StepHypRef Expression
1 nfcv 2489 . . . 4 yA
2 nfab1 2491 . . . 4 y{y φ}
31, 2nfiin 3996 . . 3 yx A {y φ}
4 nfab1 2491 . . 3 y{y x A φ}
53, 4cleqf 2513 . 2 (x A {y φ} = {y x A φ} ↔ y(y x A {y φ} ↔ y {y x A φ}))
6 abid 2341 . . . 4 (y {y φ} ↔ φ)
76ralbii 2638 . . 3 (x A y {y φ} ↔ x A φ)
8 vex 2862 . . . 4 y V
9 eliin 3974 . . . 4 (y V → (y x A {y φ} ↔ x A y {y φ}))
108, 9ax-mp 5 . . 3 (y x A {y φ} ↔ x A y {y φ})
11 abid 2341 . . 3 (y {y x A φ} ↔ x A φ)
127, 10, 113bitr4i 268 . 2 (y x A {y φ} ↔ y {y x A φ})
135, 12mpgbir 1550 1 x A {y φ} = {y x A φ}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  Vcvv 2859  ∩ciin 3970 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-iin 3972 This theorem is referenced by:  iinrab  4028
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