Theorem 19.9t 1779

Description: A closed version of 19.9 1783. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.)

Assertion

Ref	Expression
19.9t	|- ( F/xph -> (E.xph <-> ph))

Proof of Theorem 19.9t

Step	Hyp	Ref	Expression
1		df-nf 1545	. . 3 |- ( F/xph <-> A.x(ph -> A.xph))
2		19.9ht 1778	. . 3 |- (A.x(ph -> A.xph) -> (E.xph -> ph))
3	1, 2	sylbi 187	. 2 |- ( F/xph -> (E.xph -> ph))
4		19.8a 1756	. 2 |- (ph -> E.xph)
5	3, 4	impbid1 194	1 |- ( F/xph -> (E.xph <-> ph))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  19.9h  1780  19.9d  1782  19.9OLD  1784  19.21t  1795  19.23t  1800  19.23tOLD  1819  vtoclegft  2927