Theorem abbi2i 2353

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 2347	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		abbiri.1	. 2 |- ( x e. A <-> ph )
3	1, 2	mpgbir 1506	1 |- A = { x | ph }

Syntax hints:   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234

This theorem is referenced by:  abid2  2361  cbvralcsf  3210  cbvrexcsf  3211  cbvreucsf  3212  cbvrabcsf  3213  symdifxor  3497  dfpr2  3727  dftp2  3757  0iin  4069  pwpwab  4098  epse  4485  fv3  5716  fo1st  6385  fo2nd  6386  xp2  6401  tfrlem3  6576  tfr1onlem3  6603  mapsn  6966  ixpconstg  6983  ixp0x  7002  nnzrab  9651  nn0zrab  9652