Theorem abbi2i 2202

Description: Equality of a class variable and a class abstraction (inference form). (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 2196	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		abbiri.1	. 2 |- ( x e. A <-> ph )
3	1, 2	mpgbir 1387	1 |- A = { x | ph }

Syntax hints:   <-> wb 103   = wceq 1289   e. wcel 1438   {cab 2074

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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084

This theorem is referenced by:  abid2  2208  cbvralcsf  2990  cbvrexcsf  2991  cbvreucsf  2992  cbvrabcsf  2993  symdifxor  3265  dfnul2  3288  dfpr2  3465  dftp2  3491  0iin  3788  pwpwab  3816  epse  4169  fv3  5328  fo1st  5928  fo2nd  5929  xp2  5943  tfrlem3  6076  tfr1onlem3  6103  mapsn  6447  nnzrab  8774  nn0zrab  8775