Theorem ax17e 1618

Description: A rephrasing of ax-17 1616 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)

Assertion

Ref	Expression
ax17e	|- (E.xph -> ph)

Distinct variable group:   ph,x

Proof of Theorem ax17e

Step	Hyp	Ref	Expression
1		df-ex 1542	. 2 |- (E.xph <-> -. A.x -. ph)
2		ax-17 1616	. . 3 |- (-. ph -> A.x -. ph)
3	2	con1i 121	. 2 |- (-. A.x -. ph -> ph)
4	1, 3	sylbi 187	1 |- (E.xph -> ph)

This proof depends on syntax axioms:  -. wn 3   -> wi 4  A.wal 1540  E.wex 1541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-17 1616

This proof depends on definitions:  df-bi 177  df-ex 1542

This theorem is used by:  exlimiv  1634  exlimdv  1636  19.9v  1664  equid  1676