Theorem abbi2i 2193

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 2187	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		abbiri.1	. 2 |- ( x e. A <-> ph )
3	1, 2	mpgbir 1382	1 |- A = { x | ph }

Syntax hints:   <-> wb 103   = wceq 1284   e. wcel 1433   {cab 2067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077

This theorem is referenced by:  abid2  2199  cbvralcsf  2964  cbvrexcsf  2965  cbvreucsf  2966  cbvrabcsf  2967  symdifxor  3230  dfnul2  3253  dfpr2  3417  dftp2  3441  0iin  3736  epse  4097  fv3  5218  fo1st  5804  fo2nd  5805  xp2  5819  tfrlem3  5949  nnzrab  8375  nn0zrab  8376