Theorem 19.23t 1670

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 1592	. . 3 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )
2		19.9t 1635	. . . 4 |- ( F/ x ps -> ( E. x ps <-> ps ) )
3	2	biimpd 143	. . 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 1511	. . . 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 1669	. . 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 128	1 |- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   F/wnf 1453   E.wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  19.23  1671  r19.23t  2577  ceqsalt  2756  vtoclgft  2780  sbciegft  2985