Theorem eqabdv 2369

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 1939	. . 3 |- ( ph -> ( [ y / x ] x e. A <-> [ y / x ] ps ) )
3		clelsb1 2343	. . . 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 2225	. . 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 2236	1 |- ( ph -> A = { x | ps } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   [wsb 1815   e. wcel 2209   {cab 2224

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234

This theorem is referenced by:  eqabcdv  2370  eqabi  2371  mapsnd  6960  wrdval  11285  wrdnval  11313  dfrhm2  14434  rspsn  14843