Theorem abbi2i 2344

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 2338	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		abbiri.1	. 2 |- ( x e. A <-> ph )
3	1, 2	mpgbir 1499	1 |- A = { x | ph }

Syntax hints:   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225

This theorem is referenced by:  abid2  2350  cbvralcsf  3187  cbvrexcsf  3188  cbvreucsf  3189  cbvrabcsf  3190  symdifxor  3470  dfnul2  3493  dfpr2  3685  dftp2  3715  0iin  4024  pwpwab  4053  epse  4433  fv3  5650  fo1st  6303  fo2nd  6304  xp2  6319  tfrlem3  6457  tfr1onlem3  6484  mapsn  6837  ixpconstg  6854  ixp0x  6873  nnzrab  9470  nn0zrab  9471