Theorem 19.23t 1701

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 1623	. . 3 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )
2		19.9t 1666	. . . 4 |- ( F/ x ps -> ( E. x ps <-> ps ) )
3	2	biimpd 144	. . 3 |- ( F/ x ps -> ( E. x ps -> ps ) )
4	1, 3	syl9r 73	. 2 |- ( F/ x ps -> ( A. x ( ph -> ps ) -> ( E. x ph -> ps ) ) )
5		nfr 1542	. . . 4 |- ( F/ x ps -> ( ps -> A. x ps ) )
6	5	imim2d 54	. . 3 |- ( F/ x ps -> ( ( E. x ph -> ps ) -> ( E. x ph -> A. x ps ) ) )
7		19.38 1700	. . 3 |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
8	6, 7	syl6 33	. 2 |- ( F/ x ps -> ( ( E. x ph -> ps ) -> A. x ( ph -> ps ) ) )
9	4, 8	impbid 129	1 |- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   F/wnf 1484   E.wex 1516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485

This theorem is referenced by:  19.23  1702  r19.23t  2615  ceqsalt  2803  vtoclgft  2828  sbciegft  3036