Theorem abeq1i 2308

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 2184	. 2 |- ( x e. { x | ph } <-> ph )
2		abeqri.1	. . 3 |- { x | ph } = A
3	2	eleq2i 2263	. 2 |- ( x e. { x | ph } <-> x e. A )
4	1, 3	bitr3i 186	1 |- ( ph <-> x e. A )

Syntax hints:   <-> wb 105   = wceq 1364   e. wcel 2167   {cab 2182

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192

This theorem is referenced by: (None)