Theorem abbi2i 2320

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 2314	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		abbiri.1	. 2 |- ( x e. A <-> ph )
3	1, 2	mpgbir 1476	1 |- A = { x | ph }

Syntax hints:   <-> wb 105   = wceq 1373   e. wcel 2176   {cab 2191

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201

This theorem is referenced by:  abid2  2326  cbvralcsf  3156  cbvrexcsf  3157  cbvreucsf  3158  cbvrabcsf  3159  symdifxor  3439  dfnul2  3462  dfpr2  3652  dftp2  3682  0iin  3986  pwpwab  4015  epse  4389  fv3  5599  fo1st  6243  fo2nd  6244  xp2  6259  tfrlem3  6397  tfr1onlem3  6424  mapsn  6777  ixpconstg  6794  ixp0x  6813  nnzrab  9396  nn0zrab  9397