Theorem bj-exalimi 32612

Description: An inference for distributing quantifiers over a double implication. (Almost) the general statement that speimfw 1876 proves. (Contributed by BJ, 29-Sep-2019.)

Hypothesis

Ref	Expression
bj-exalimi.1	|- ( ph -> ( ps -> ch ) )

Assertion

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

Proof of Theorem bj-exalimi

Step	Hyp	Ref	Expression
1		bj-exalimi.1	. . . 4 |- ( ph -> ( ps -> ch ) )
2	1	com12 32	. . 3 |- ( ps -> ( ph -> ch ) )
3	2	aleximi 1759	. 2 |- ( A. x ps -> ( E. x ph -> E. x ch ) )
4	3	com12 32	1 |- ( E. x ph -> ( A. x ps -> E. x ch ) )

Syntax hints:   -> 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)