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Theorem clabel 2145
 Description: Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
Assertion
Ref Expression
clabel ({xφ} Ay(y A x(x yφ)))
Distinct variable groups:   y,A   φ,y   x,y
Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem clabel
StepHypRef Expression
1 df-clel 2018 . 2 ({xφ} Ay(y = {xφ} y A))
2 abeq2 2128 . . . 4 (y = {xφ} ↔ x(x yφ))
32anbi2ci 435 . . 3 ((y = {xφ} y A) ↔ (y A x(x yφ)))
43exbii 1478 . 2 (y(y = {xφ} y A) ↔ y(y A x(x yφ)))
51, 4bitri 173 1 ({xφ} Ay(y A x(x yφ)))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∀wal 1226   = wceq 1228  ∃wex 1362   ∈ wcel 1374  {cab 2008 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-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  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-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018 This theorem is referenced by: (None)
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