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Theorem clelab 2473
 Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
Assertion
Ref Expression
clelab (A {x φ} ↔ x(x = A φ))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem clelab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-clab 2340 . . . 4 (y {x φ} ↔ [y / x]φ)
21anbi2i 675 . . 3 ((y = A y {x φ}) ↔ (y = A [y / x]φ))
32exbii 1582 . 2 (y(y = A y {x φ}) ↔ y(y = A [y / x]φ))
4 df-clel 2349 . 2 (A {x φ} ↔ y(y = A y {x φ}))
5 nfv 1619 . . 3 y(x = A φ)
6 nfv 1619 . . . 4 x y = A
7 nfs1v 2106 . . . 4 x[y / x]φ
86, 7nfan 1824 . . 3 x(y = A [y / x]φ)
9 eqeq1 2359 . . . 4 (x = y → (x = Ay = A))
10 sbequ12 1919 . . . 4 (x = y → (φ ↔ [y / x]φ))
119, 10anbi12d 691 . . 3 (x = y → ((x = A φ) ↔ (y = A [y / x]φ)))
125, 8, 11cbvex 1985 . 2 (x(x = A φ) ↔ y(y = A [y / x]φ))
133, 4, 123bitr4i 268 1 (A {x φ} ↔ x(x = A φ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ 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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