Theorem 19.23t 2079

Description: Closed form of Theorem 1977.23 of [Margaris] p. 90. See 19.23 2080. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.)

Assertion

Ref	Expression
19.23t	|- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )

Proof of Theorem 19.23t

Step	Hyp	Ref	Expression
1		nfnt 1782	. . 3 |- ( F/ x ps -> F/ x -. ps )
2		19.21t 2073	. . 3 |- ( F/ x -. ps -> ( A. x ( -. ps -> -. ph ) <-> ( -. ps -> A. x -. ph ) ) )
3	1, 2	syl 17	. 2 |- ( F/ x ps -> ( A. x ( -. ps -> -. ph ) <-> ( -. ps -> A. x -. ph ) ) )
4		con34b 306	. . 3 |- ( ( ph -> ps ) <-> ( -. ps -> -. ph ) )
5	4	albii 1747	. 2 |- ( A. x ( ph -> ps ) <-> A. x ( -. ps -> -. ph ) )
6		eximal 1707	. 2 |- ( ( E. x ph -> ps ) <-> ( -. ps -> A. x -. ph ) )
7	3, 5, 6	3bitr4g 303	1 |- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708

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-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by:  19.23  2080  axie2  2597  r19.23t  3021  ceqsalt  3228  vtoclgft  3254  vtoclgftOLD  3255  sbciegft  3466  bj-ceqsalt0  32873  bj-ceqsalt1  32874  wl-equsald  33325