Theorem nabbi 2896

Description: Not equivalent wff's correspond to not equal class abstractions. (Contributed by AV, 7-Apr-2019.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)

Assertion

Ref	Expression
nabbi	|- ( E. x ( ph <-> -. ps ) <-> { x | ph } =/= { x | ps } )

Proof of Theorem nabbi

Step	Hyp	Ref	Expression
1		df-ne 2795	. 2 |- ( { x | ph } =/= { x | ps } <-> -. { x | ph } = { x | ps } )
2		exnal 1754	. . . 4 |- ( E. x -. ( ph <-> ps ) <-> -. A. x ( ph <-> ps ) )
3		xor3 372	. . . . 5 |- ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) )
4	3	exbii 1774	. . . 4 |- ( E. x -. ( ph <-> ps ) <-> E. x ( ph <-> -. ps ) )
5	2, 4	bitr3i 266	. . 3 |- ( -. A. x ( ph <-> ps ) <-> E. x ( ph <-> -. ps ) )
6		abbi 2737	. . 3 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )
7	5, 6	xchnxbi 322	. 2 |- ( -. { x | ph } = { x | ps } <-> E. x ( ph <-> -. ps ) )
8	1, 7	bitr2i 265	1 |- ( E. x ( ph <-> -. ps ) <-> { x | ph } =/= { x | ps } )

Syntax hints:   -. wn 3   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704   {cab 2608   =/= wne 2794

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  df-ne 2795

This theorem is referenced by:  suppvalbr  7299