Theorem exlimd2 1609

Description: Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1610 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.)

Hypotheses

Ref	Expression
exlimd2.1	|- ( ph -> A. x ph )
exlimd2.2	|- ( ph -> ( ch -> A. x ch ) )
exlimd2.3	|- ( ph -> ( ps -> ch ) )

Assertion

Ref	Expression
exlimd2	|- ( ph -> ( E. x ps -> ch ) )

Proof of Theorem exlimd2

Step	Hyp	Ref	Expression
1		exlimd2.1	. . 3 |- ( ph -> A. x ph )
2		exlimd2.2	. . 3 |- ( ph -> ( ch -> A. x ch ) )
3	1, 2	alrimih 1483	. 2 |- ( ph -> A. x ( ch -> A. x ch ) )
4		exlimd2.3	. . 3 |- ( ph -> ( ps -> ch ) )
5	1, 4	alrimih 1483	. 2 |- ( ph -> A. x ( ps -> ch ) )
6		19.23ht 1511	. . 3 |- ( A. x ( ch -> A. x ch ) -> ( A. x ( ps -> ch ) <-> ( E. x ps -> ch ) ) )
7	6	biimpd 144	. 2 |- ( A. x ( ch -> A. x ch ) -> ( A. x ( ps -> ch ) -> ( E. x ps -> ch ) ) )
8	3, 5, 7	sylc 62	1 |- ( ph -> ( E. x ps -> ch ) )

Syntax hints:   -> wi 4   A.wal 1362   E.wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-5 1461  ax-gen 1463  ax-ie2 1508

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  equsexd  1743  cbvexdh  1941