Theorem abbi1dv 2743

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
abbi1dv.1	|- ( ph -> ( ps <-> x e. A ) )

Assertion

Ref	Expression
abbi1dv	|- ( ph -> { x | ps } = A )

Distinct variable groups:   x, A   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem abbi1dv

Step	Hyp	Ref	Expression
1		abbi1dv.1	. . . 4 |- ( ph -> ( ps <-> x e. A ) )
2	1	bicomd 213	. . 3 |- ( ph -> ( x e. A <-> ps ) )
3	2	abbi2dv 2742	. 2 |- ( ph -> A = { x | ps } )
4	3	eqcomd 2628	1 |- ( ph -> { x | ps } = A )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618

This theorem is referenced by:  abidnf  3375  csbtt  3544  csbie2g  3564  csbvarg  4003  iinxsng  4600  predep  5706  enfin2i  9143  fin1a2lem11  9232  hashf1  13241  shftuz  13809  psrbaglefi  19372  vmappw  24842  hdmap1fval  37086  hdmapfval  37119  hgmapfval  37178