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Theorem abeq2i 2145
 Description: Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)
Hypothesis
Ref Expression
abeqi.1 A = {xφ}
Assertion
Ref Expression
abeq2i (x Aφ)

Proof of Theorem abeq2i
StepHypRef Expression
1 abeqi.1 . . 3 A = {xφ}
21eleq2i 2101 . 2 (x Ax {xφ})
3 abid 2025 . 2 (x {xφ} ↔ φ)
42, 3bitri 173 1 (x Aφ)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = 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-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033 This theorem is referenced by:  rabid  2479  vex  2554  csbco  2855  csbnestgf  2892  pwss  3366  elsn  3382  snsspw  3526  iunpw  4177  ordon  4178  funcnv3  4904  tfrlem4  5870  tfrlem8  5875  tfrlem9  5876  tfrlemibxssdm  5882  1idprl  6565  1idpru  6566  recexprlem1ssl  6604  recexprlem1ssu  6605  recexprlemss1l  6606  recexprlemss1u  6607
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