Theorem abbi 2576

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 2451	. . 3 |- ( y e. { x | ph } -> A. x y e. { x | ph } )
2		hbab1 2451	. . 3 |- ( y e. { x | ps } -> A. x y e. { x | ps } )
3	1, 2	cleqh 2563	. 2 |- ( { x | ph } = { x | ps } <-> A. x ( x e. { x | ph } <-> x e. { x | ps } ) )
4		abid 2450	. . . 4 |- ( x e. { x | ph } <-> ph )
5		abid 2450	. . . 4 |- ( x e. { x | ps } <-> ps )
6	4, 5	bibi12i 321	. . 3 |- ( ( x e. { x | ph } <-> x e. { x | ps } ) <-> ( ph <-> ps ) )
7	6	albii 1702	. 2 |- ( A. x ( x e. { x | ph } <-> x e. { x | ps } ) <-> A. x ( ph <-> ps ) )
8	3, 7	bitr2i 258	1 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )

Syntax hints:   <-> wb 189   A.wal 1453   = wceq 1455   e. wcel 1898   {cab 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458

This theorem is referenced by:  abbii  2578  abbid  2579  nabbi  2737  rabbi  2981  sbcbi2  3328  rabeqsn  4013  iuneq12df  4316  dfiota2  5570  iotabi  5578  uniabio  5579  iotanul  5584  karden  8397  iuneq12daf  28225  bj-cleq  31601  abeq12  32445  elnev  36834  csbingVD  37322  csbsngVD  37331  csbxpgVD  37332  csbrngVD  37334  csbunigVD  37336  csbfv12gALTVD  37337