Theorem eqabdv 2336

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 1909	. . 3 |- ( ph -> ( [ y / x ] x e. A <-> [ y / x ] ps ) )
3		clelsb1 2312	. . . 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 2194	. . 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 2205	1 |- ( ph -> A = { x | ps } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   [wsb 1786   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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203

This theorem is referenced by:  wrdval  11034  wrdnval  11061  dfrhm2  14031  rspsn  14411