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Theorem abeq1 2144
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 2143 . 2 (A = {xφ} ↔ x(x Aφ))
2 eqcom 2039 . 2 ({xφ} = AA = {xφ})
3 bicom 128 . . 3 ((φx A) ↔ (x Aφ))
43albii 1356 . 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 1240   = wceq 1242   wcel 1390  {cab 2023
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033
This theorem is referenced by:  abbi1dv  2154  disj  3262  euabsn2  3430  dm0rn0  4495  dffo3  5257
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