Theorem 19.21t 2073

Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2075. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.)

Assertion

Ref	Expression
19.21t	|- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )

Proof of Theorem 19.21t

Step	Hyp	Ref	Expression
1		19.38a 1767	. 2 |- ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> A. x ( ph -> ps ) ) )
2		19.9t 2071	. . 3 |- ( F/ x ph -> ( E. x ph <-> ph ) )
3	2	imbi1d 331	. 2 |- ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> ( ph -> A. x ps ) ) )
4	1, 3	bitr3d 270	1 |- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )

Syntax hints:   -> 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-ex 1705  df-nf 1710

This theorem is referenced by:  19.21  2075  stdpc5OLD  2077  19.23t  2079  sbal1  2460  sbal2  2461  r19.21t  2955  ceqsalt  3228  sbciegft  3466  bj-ceqsalt0  32873  bj-ceqsalt1  32874  wl-sbhbt  33335  wl-2sb6d  33341  wl-sbalnae  33345  ax12indalem  34230  ax12inda2ALT  34231