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Theorem clabel 2474
 Description: Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
Assertion
Ref Expression
clabel ({x φ} 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 2349 . 2 ({x φ} Ay(y = {x φ} y A))
2 abeq2 2458 . . . 4 (y = {x φ} ↔ x(x yφ))
32anbi2ci 677 . . 3 ((y = {x φ} y A) ↔ (y A x(x yφ)))
43exbii 1582 . 2 (y(y = {x φ} y A) ↔ y(y A x(x yφ)))
51, 4bitri 240 1 ({x φ} Ay(y A x(x yφ)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349 This theorem is referenced by: (None)
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