Theorem abbi2i 2255

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 2249	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		abbiri.1	. 2 |- ( x e. A <-> ph )
3	1, 2	mpgbir 1430	1 |- A = { x | ph }

Syntax hints:   <-> wb 104   = wceq 1332   e. wcel 1481   {cab 2126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136

This theorem is referenced by:  abid2  2261  cbvralcsf  3067  cbvrexcsf  3068  cbvreucsf  3069  cbvrabcsf  3070  symdifxor  3347  dfnul2  3370  dfpr2  3551  dftp2  3580  0iin  3879  pwpwab  3908  epse  4272  fv3  5452  fo1st  6063  fo2nd  6064  xp2  6079  tfrlem3  6216  tfr1onlem3  6243  mapsn  6592  ixpconstg  6609  ixp0x  6628  nnzrab  9102  nn0zrab  9103