Theorem eqabcdv 2364

Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)

Hypothesis

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

Assertion

Ref	Expression
eqabcdv	|- ( ph -> { x | ps } = A )

Distinct variable groups:   x, A   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem eqabcdv

Step	Hyp	Ref	Expression
1		eqabcdv.1	. . . 4 |- ( ph -> ( ps <-> x e. A ) )
2	1	bicomd 141	. . 3 |- ( ph -> ( x e. A <-> ps ) )
3	2	eqabdv 2363	. 2 |- ( ph -> A = { x | ps } )
4	3	eqcomd 2238	1 |- ( ph -> { x | ps } = A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   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-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228

This theorem is referenced by: (None)