Theorem abbi2i 2347

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 2341	. 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 2203   {cab 2218

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 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228

This theorem is referenced by:  abid2  2355  cbvralcsf  3200  cbvrexcsf  3201  cbvreucsf  3202  cbvrabcsf  3203  symdifxor  3486  dfnul2  3509  dfpr2  3707  dftp2  3737  0iin  4049  pwpwab  4078  epse  4462  fv3  5692  fo1st  6350  fo2nd  6351  xp2  6366  tfrlem3  6541  tfr1onlem3  6568  mapsn  6924  ixpconstg  6941  ixp0x  6960  nnzrab  9597  nn0zrab  9598