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Theorem iinab 3692
 Description: Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
Assertion
Ref Expression
iinab x A {yφ} = {yx 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 2160 . . . 4 yA
2 nfab1 2162 . . . 4 y{yφ}
31, 2nfiinxy 3658 . . 3 y x A {yφ}
4 nfab1 2162 . . 3 y{yx A φ}
53, 4cleqf 2183 . 2 ( x A {yφ} = {yx A φ} ↔ y(y x A {yφ} ↔ y {yx A φ}))
6 abid 2010 . . . 4 (y {yφ} ↔ φ)
76ralbii 2308 . . 3 (x A y {yφ} ↔ x A φ)
8 vex 2538 . . . 4 y V
9 eliin 3636 . . . 4 (y V → (y x A {yφ} ↔ x A y {yφ}))
108, 9ax-mp 7 . . 3 (y x A {yφ} ↔ x A y {yφ})
11 abid 2010 . . 3 (y {yx A φ} ↔ x A φ)
127, 10, 113bitr4i 201 . 2 (y x A {yφ} ↔ y {yx A φ})
135, 12mpgbir 1322 1 x A {yφ} = {yx A φ}
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1228   ∈ wcel 1374  {cab 2008  ∀wral 2284  Vcvv 2535  ∩ ciin 3632 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-iin 3634 This theorem is referenced by:  iinrabm  3693
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