Theorem abbi 2548

Description: Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)

Assertion

Ref	Expression
abbi	|- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )

Proof of Theorem abbi

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbab1 2410	. . 3 |- ( y e. { x | ph } -> A. x y e. { x | ph } )
2		hbab1 2410	. . 3 |- ( y e. { x | ps } -> A. x y e. { x | ps } )
3	1, 2	cleqh 2532	. 2 |- ( { x | ph } = { x | ps } <-> A. x ( x e. { x | ph } <-> x e. { x | ps } ) )
4		abid 2409	. . . 4 |- ( x e. { x | ph } <-> ph )
5		abid 2409	. . . 4 |- ( x e. { x | ps } <-> ps )
6	4, 5	bibi12i 316	. . 3 |- ( ( x e. { x | ph } <-> x e. { x | ps } ) <-> ( ph <-> ps ) )
7	6	albii 1685	. 2 |- ( A. x ( x e. { x | ph } <-> x e. { x | ps } ) <-> A. x ( ph <-> ps ) )
8	3, 7	bitr2i 253	1 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )

Syntax hints:   <-> wb 187   A.wal 1435   = wceq 1437   e. wcel 1872   {cab 2407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417

This theorem is referenced by:  abbii  2551  abbid  2552  nabbi  2754  nabbiOLD  2755  rabbi  3004  sbcbi2  3348  iuneq12df  4323  dfiota2  5566  iotabi  5574  uniabio  5575  iotanul  5580  karden  8374  iuneq12daf  28172  bj-cleq  31523  abeq12  32363  elnev  36759  csbingVD  37254  csbsngVD  37263  csbxpgVD  37264  csbrngVD  37266  csbunigVD  37268  csbfv12gALTVD  37269  rabeqsn  39733