Theorem abeq1i 2190

Description: Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)

Hypothesis

Ref	Expression
abeqri.1	|- { x | ph } = A

Assertion

Ref	Expression
abeq1i	|- ( ph <-> x e. A )

Proof of Theorem abeq1i

Step	Hyp	Ref	Expression
1		abid 2069	. 2 |- ( x e. { x | ph } <-> ph )
2		abeqri.1	. . 3 |- { x | ph } = A
3	2	eleq2i 2145	. 2 |- ( x e. { x | ph } <-> x e. A )
4	1, 3	bitr3i 184	1 |- ( ph <-> x e. A )

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-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

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

This theorem is referenced by: (None)