Theorem abbi2i 2149

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 2143	. 2 |- (A = {x | ph } <-> A.x(x e. A <-> ph))
2		abbiri.1	. 2 |- (x e. A <-> ph)
3	1, 2	mpgbir 1339	1 |- A = {x | ph }

Syntax hints:   <-> wb 98   = wceq 1242   e. wcel 1390   {cab 2023

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033

This theorem is referenced by:  abid2  2155  cbvralcsf  2902  cbvrexcsf  2903  cbvreucsf  2904  cbvrabcsf  2905  symdifxor  3197  dfnul2  3220  dfpr2  3383  dftp2  3410  0iin  3706  epse  4064  fv3  5140  fo1st  5726  fo2nd  5727  xp2  5741  tfrlem3  5867  nnzrab  8025  nn0zrab  8026