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Theorem abeq1 2129
Description: Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
Assertion
Ref Expression
abeq1 ({xφ} = Ax(φx A))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem abeq1
StepHypRef Expression
1 abeq2 2128 . 2 (A = {xφ} ↔ x(x Aφ))
2 eqcom 2024 . 2 ({xφ} = AA = {xφ})
3 bicom 128 . . 3 ((φx A) ↔ (x Aφ))
43albii 1339 . 2 (x(φx A) ↔ x(x Aφ))
51, 2, 43bitr4i 201 1 ({xφ} = Ax(φx A))
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1226   = wceq 1228   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:  abbi1dv  2139  disj  3245  euabsn2  3413  dm0rn0  4479  dffo3  5239
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