Theorem 19.23t 1800

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/xps -> (A.x(ph -> ps) <-> (E.xph -> ps)))

Proof of Theorem 19.23t

Step	Hyp	Ref	Expression
1		exim 1575	. . 3 |- (A.x(ph -> ps) -> (E.xph -> E.xps))
2		19.9t 1779	. . . 4 |- ( F/xps -> (E.xps <-> ps))
3	2	biimpd 198	. . 3 |- ( F/xps -> (E.xps -> ps))
4	1, 3	syl9r 67	. 2 |- ( F/xps -> (A.x(ph -> ps) -> (E.xph -> ps)))
5		nfr 1761	. . . 4 |- ( F/xps -> (ps -> A.xps))
6	5	imim2d 48	. . 3 |- ( F/xps -> ((E.xph -> ps) -> (E.xph -> A.xps)))
7		19.38 1794	. . 3 |- ((E.xph -> A.xps) -> A.x(ph -> ps))
8	6, 7	syl6 29	. 2 |- ( F/xps -> ((E.xph -> ps) -> A.x(ph -> ps)))
9	4, 8	impbid 183	1 |- ( F/xps -> (A.x(ph -> ps) <-> (E.xph -> ps)))

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:  19.23  1801  sbft  2025  axie2  2329  r19.23t  2729  ceqsalt  2882  vtoclgft  2906  sbciegft  3077