Theorem eqabi 2365

Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 26-May-1993.) (Revised by Wolf Lammen, 6-May-2023.)

Hypothesis

Ref	Expression
eqabi.1	|- ( x e. A <-> ph )

Assertion

Ref	Expression
eqabi	|- A = { x | ph }

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem eqabi

Step	Hyp	Ref	Expression
1		eqabi.1	. . . 4 |- ( x e. A <-> ph )
2	1	a1i 9	. . 3 |- ( T. -> ( x e. A <-> ph ) )
3	2	eqabdv 2363	. 2 |- ( T. -> A = { x | ph } )
4	3	mptru 1407	1 |- A = { x | ph }

Syntax hints:   <-> wb 105   = wceq 1398   T. wtru 1399   e. wcel 2203   {cab 2218

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228

This theorem is referenced by:  abid1  2366