Theorem dfnf5 3952

Description: Characterization of non-freeness in a formula in terms of its extension. (Contributed by BJ, 19-Mar-2021.)

Assertion

Ref	Expression
dfnf5	|- ( F/ x ph <-> ( { x | ph } = (/) \/ { x | ph } = _V ) )

Proof of Theorem dfnf5

Step	Hyp	Ref	Expression
1		df-ex 1705	. . . 4 |- ( E. x ph <-> -. A. x -. ph )
2	1	imbi1i 339	. . 3 |- ( ( E. x ph -> A. x ph ) <-> ( -. A. x -. ph -> A. x ph ) )
3		pm4.64 387	. . 3 |- ( ( -. A. x -. ph -> A. x ph ) <-> ( A. x -. ph \/ A. x ph ) )
4	2, 3	bitri 264	. 2 |- ( ( E. x ph -> A. x ph ) <-> ( A. x -. ph \/ A. x ph ) )
5		df-nf 1710	. 2 |- ( F/ x ph <-> ( E. x ph -> A. x ph ) )
6		ab0 3951	. . 3 |- ( { x | ph } = (/) <-> A. x -. ph )
7		abv 3206	. . 3 |- ( { x | ph } = _V <-> A. x ph )
8	6, 7	orbi12i 543	. 2 |- ( ( { x | ph } = (/) \/ { x | ph } = _V ) <-> ( A. x -. ph \/ A. x ph ) )
9	4, 5, 8	3bitr4i 292	1 |- ( F/ x ph <-> ( { x | ph } = (/) \/ { x | ph } = _V ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   A.wal 1481   = wceq 1483   E.wex 1704   F/wnf 1708   {cab 2608   _Vcvv 3200   (/)c0 3915

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-nfc 2753  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  ab0orv  3953