Theorem abbi 2737

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 2611	. . 3 |- ( y e. { x | ph } -> A. x y e. { x | ph } )
2		hbab1 2611	. . 3 |- ( y e. { x | ps } -> A. x y e. { x | ps } )
3	1, 2	cleqh 2724	. 2 |- ( { x | ph } = { x | ps } <-> A. x ( x e. { x | ph } <-> x e. { x | ps } ) )
4		abid 2610	. . . 4 |- ( x e. { x | ph } <-> ph )
5		abid 2610	. . . 4 |- ( x e. { x | ps } <-> ps )
6	4, 5	bibi12i 329	. . 3 |- ( ( x e. { x | ph } <-> x e. { x | ps } ) <-> ( ph <-> ps ) )
7	6	albii 1747	. 2 |- ( A. x ( x e. { x | ph } <-> x e. { x | ps } ) <-> A. x ( ph <-> ps ) )
8	3, 7	bitr2i 265	1 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )

Syntax hints:   <-> wb 196   A.wal 1481   = 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:  abbii  2739  abbid  2740  nabbi  2896  rabbi  3120  sbcbi2  3484  rabeqsn  4214  iuneq12df  4544  dfiota2  5852  iotabi  5860  uniabio  5861  iotanul  5866  karden  8758  iuneq12daf  29373  bj-cleq  32949  abeq12  33964  elnev  38639  csbingVD  39120  csbsngVD  39129  csbxpgVD  39130  csbrngVD  39132  csbunigVD  39134  csbfv12gALTVD  39135