Theorem abbi2i 2194

Description: Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
abbiri.1	|- ( x e. A <-> ph )

Assertion

Ref	Expression
abbi2i	|- A = { x | ph }

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem abbi2i

Step	Hyp	Ref	Expression
1		abeq2 2188	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		abbiri.1	. 2 |- ( x e. A <-> ph )
3	1, 2	mpgbir 1383	1 |- A = { x | ph }

Syntax hints:   <-> wb 103   = wceq 1285   e. wcel 1434   {cab 2068

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078

This theorem is referenced by:  abid2  2200  cbvralcsf  2965  cbvrexcsf  2966  cbvreucsf  2967  cbvrabcsf  2968  symdifxor  3231  dfnul2  3254  dfpr2  3419  dftp2  3443  0iin  3738  epse  4099  fv3  5223  fo1st  5809  fo2nd  5810  xp2  5824  tfrlem3  5954  tfr1onlem3  5981  nnzrab  8445  nn0zrab  8446