Theorem abbi2i 2254

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 2248	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		abbiri.1	. 2 |- ( x e. A <-> ph )
3	1, 2	mpgbir 1429	1 |- A = { x | ph }

Syntax hints:   <-> wb 104   = wceq 1331   e. wcel 1480   {cab 2125

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135

This theorem is referenced by:  abid2  2260  cbvralcsf  3062  cbvrexcsf  3063  cbvreucsf  3064  cbvrabcsf  3065  symdifxor  3342  dfnul2  3365  dfpr2  3546  dftp2  3572  0iin  3871  pwpwab  3900  epse  4264  fv3  5444  fo1st  6055  fo2nd  6056  xp2  6071  tfrlem3  6208  tfr1onlem3  6235  mapsn  6584  ixpconstg  6601  ixp0x  6620  nnzrab  9078  nn0zrab  9079