Theorem eqabdv 2358

Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Revised by Wolf Lammen, 6-May-2023.)

Hypothesis

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

Assertion

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

Distinct variable groups:   x, A   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem eqabdv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqabdv.1	. . . 4 |- ( ph -> ( x e. A <-> ps ) )
2	1	sbbidv 1931	. . 3 |- ( ph -> ( [ y / x ] x e. A <-> [ y / x ] ps ) )
3		clelsb1 2334	. . . 4 |- ( [ y / x ] x e. A <-> y e. A )
4	3	bicomi 132	. . 3 |- ( y e. A <-> [ y / x ] x e. A )
5		df-clab 2216	. . 3 |- ( y e. { x | ps } <-> [ y / x ] ps )
6	2, 4, 5	3bitr4g 223	. 2 |- ( ph -> ( y e. A <-> y e. { x | ps } ) )
7	6	eqrdv 2227	1 |- ( ph -> A = { x | ps } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   [wsb 1808   e. wcel 2200   {cab 2215

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225

This theorem is referenced by:  wrdval  11074  wrdnval  11102  dfrhm2  14118  rspsn  14498