Theorem abeq1i 2319

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 2195	. 2 |- ( x e. { x | ph } <-> ph )
2		abeqri.1	. . 3 |- { x | ph } = A
3	2	eleq2i 2274	. 2 |- ( x e. { x | ph } <-> x e. A )
4	1, 3	bitr3i 186	1 |- ( ph <-> x e. A )

Syntax hints:   <-> wb 105   = wceq 1373   e. wcel 2178   {cab 2193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203

This theorem is referenced by: (None)