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Theorem elintab 3600
 Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
inteqab.1 A V
Assertion
Ref Expression
elintab (A {xφ} ↔ x(φA x))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem elintab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 inteqab.1 . . 3 A V
21elint 3595 . 2 (A {xφ} ↔ y(y {xφ} → A y))
3 nfsab1 2012 . . . 4 x y {xφ}
4 nfv 1402 . . . 4 x A y
53, 4nfim 1446 . . 3 x(y {xφ} → A y)
6 nfv 1402 . . 3 y(φA x)
7 eleq1 2082 . . . . 5 (y = x → (y {xφ} ↔ x {xφ}))
8 abid 2010 . . . . 5 (x {xφ} ↔ φ)
97, 8syl6bb 185 . . . 4 (y = x → (y {xφ} ↔ φ))
10 eleq2 2083 . . . 4 (y = x → (A yA x))
119, 10imbi12d 223 . . 3 (y = x → ((y {xφ} → A y) ↔ (φA x)))
125, 6, 11cbval 1619 . 2 (y(y {xφ} → A y) ↔ x(φA x))
132, 12bitri 173 1 (A {xφ} ↔ x(φA x))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1226   ∈ wcel 1374  {cab 2008  Vcvv 2535  ∩ cint 3589 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-v 2537  df-int 3590 This theorem is referenced by:  elintrab  3601  intmin4  3617  intab  3618  intid  3934
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