Theorem 19.23t 1567

Description: Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

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		exim 1490	. . 3 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )
2		19.9t 1533	. . . 4 |- ( F/ x ps -> ( E. x ps <-> ps ) )
3	2	biimpd 132	. . 3 |- ( F/ x ps -> ( E. x ps -> ps ) )
4	1, 3	syl9r 67	. 2 |- ( F/ x ps -> ( A. x ( ph -> ps ) -> ( E. x ph -> ps ) ) )
5		nfr 1411	. . . 4 |- ( F/ x ps -> ( ps -> A. x ps ) )
6	5	imim2d 48	. . 3 |- ( F/ x ps -> ( ( E. x ph -> ps ) -> ( E. x ph -> A. x ps ) ) )
7		19.38 1566	. . 3 |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
8	6, 7	syl6 29	. 2 |- ( F/ x ps -> ( ( E. x ph -> ps ) -> A. x ( ph -> ps ) ) )
9	4, 8	impbid 120	1 |- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   F/wnf 1349   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  19.23  1568  r19.23t  2423  ceqsalt  2580  vtoclgft  2604  sbciegft  2793