Theorem bj-exalims 32613

Description: Distributing quantifiers over a double implication. (Almost) the general statement that spimfw 1878 proves. (Contributed by BJ, 29-Sep-2019.)

Hypothesis

Ref	Expression
bj-exalims.1	|- ( E. x ph -> ( -. ch -> A. x -. ch ) )

Assertion

Ref	Expression
bj-exalims	|- ( A. x ( ph -> ( ps -> ch ) ) -> ( E. x ph -> ( A. x ps -> ch ) ) )

Proof of Theorem bj-exalims

Step	Hyp	Ref	Expression
1		bj-exalim 32611	. 2 |- ( A. x ( ph -> ( ps -> ch ) ) -> ( E. x ph -> ( A. x ps -> E. x ch ) ) )
2		bj-exalims.1	. . . 4 |- ( E. x ph -> ( -. ch -> A. x -. ch ) )
3		eximal 1707	. . . 4 |- ( ( E. x ch -> ch ) <-> ( -. ch -> A. x -. ch ) )
4	2, 3	sylibr 224	. . 3 |- ( E. x ph -> ( E. x ch -> ch ) )
5	4	a1i 11	. 2 |- ( A. x ( ph -> ( ps -> ch ) ) -> ( E. x ph -> ( E. x ch -> ch ) ) )
6	1, 5	syldd 72	1 |- ( A. x ( ph -> ( ps -> ch ) ) -> ( E. x ph -> ( A. x ps -> ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  bj-exalimsi  32614