Theorem abbi2i 2152

Description: Equality of a class variable and a class abstraction (inference rule). (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 2146	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		abbiri.1	. 2 |- ( x e. A <-> ph )
3	1, 2	mpgbir 1342	1 |- A = { x | ph }

Syntax hints:   <-> wb 98   = wceq 1243   e. wcel 1393   {cab 2026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036

This theorem is referenced by:  abid2  2158  cbvralcsf  2908  cbvrexcsf  2909  cbvreucsf  2910  cbvrabcsf  2911  symdifxor  3203  dfnul2  3226  dfpr2  3394  dftp2  3419  0iin  3715  epse  4079  fv3  5197  fo1st  5784  fo2nd  5785  xp2  5799  tfrlem3  5926  nnzrab  8269  nn0zrab  8270