Theorem abbi2i 2349

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 2343	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		abbiri.1	. 2 |- ( x e. A <-> ph )
3	1, 2	mpgbir 1502	1 |- A = { x | ph }

Syntax hints:   <-> wb 105   = wceq 1398   e. wcel 2205   {cab 2220

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230

This theorem is referenced by:  abid2  2357  cbvralcsf  3203  cbvrexcsf  3204  cbvreucsf  3205  cbvrabcsf  3206  symdifxor  3489  dfnul2  3512  dfpr2  3710  dftp2  3740  0iin  4052  pwpwab  4081  epse  4465  fv3  5695  fo1st  6353  fo2nd  6354  xp2  6369  tfrlem3  6544  tfr1onlem3  6571  mapsn  6927  ixpconstg  6944  ixp0x  6963  nnzrab  9606  nn0zrab  9607