Theorem 19.23t 1608

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 1531	. . 3 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )
2		19.9t 1574	. . . 4 |- ( F/ x ps -> ( E. x ps <-> ps ) )
3	2	biimpd 142	. . 3 |- ( F/ x ps -> ( E. x ps -> ps ) )
4	1, 3	syl9r 72	. 2 |- ( F/ x ps -> ( A. x ( ph -> ps ) -> ( E. x ph -> ps ) ) )
5		nfr 1452	. . . 4 |- ( F/ x ps -> ( ps -> A. x ps ) )
6	5	imim2d 53	. . 3 |- ( F/ x ps -> ( ( E. x ph -> ps ) -> ( E. x ph -> A. x ps ) ) )
7		19.38 1607	. . 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 127	1 |- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1283   F/wnf 1390   E.wex 1422

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391

This theorem is referenced by:  19.23  1609  r19.23t  2468  ceqsalt  2626  vtoclgft  2650  sbciegft  2845