Theorem abeq1i 2200

Description: Equality of a class variable and a class abstraction (inference form). (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 2077	. 2 |- ( x e. { x | ph } <-> ph )
2		abeqri.1	. . 3 |- { x | ph } = A
3	2	eleq2i 2155	. 2 |- ( x e. { x | ph } <-> x e. A )
4	1, 3	bitr3i 185	1 |- ( ph <-> x e. A )

Syntax hints:   <-> wb 104   = wceq 1290   e. wcel 1439   {cab 2075

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085

This theorem is referenced by: (None)