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Theorem abeq1 2459
 Description: Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
Assertion
Ref Expression
abeq1 ({x φ} = Ax(φx A))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem abeq1
StepHypRef Expression
1 abeq2 2458 . 2 (A = {x φ} ↔ x(x Aφ))
2 eqcom 2355 . 2 ({x φ} = AA = {x φ})
3 bicom 191 . . 3 ((φx A) ↔ (x Aφ))
43albii 1566 . 2 (x(φx A) ↔ x(x Aφ))
51, 2, 43bitr4i 268 1 ({x φ} = Ax(φx A))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  abbi1dv  2469  euabsn2  3791  phidisjnn  4615  dm0rn0  4921  dffo3  5422
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