Theorem bj-exalimsi 32614

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

Hypotheses

Ref	Expression
bj-exalimsi.1	|- ( ph -> ( ps -> ch ) )
bj-exalimsi.2	|- ( E. x ph -> ( -. ch -> A. x -. ch ) )

Assertion

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

Proof of Theorem bj-exalimsi

Step	Hyp	Ref	Expression
1		bj-exalimsi.2	. . 3 |- ( E. x ph -> ( -. ch -> A. x -. ch ) )
2	1	bj-exalims 32613	. 2 |- ( A. x ( ph -> ( ps -> ch ) ) -> ( E. x ph -> ( A. x ps -> ch ) ) )
3		bj-exalimsi.1	. 2 |- ( ph -> ( ps -> ch ) )
4	2, 3	mpg 1724	1 |- ( 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: (None)