Theorem alexbii 1760

Description: Biconditional form of aleximi 1759. (Contributed by BJ, 16-Nov-2020.)

Hypothesis

Ref	Expression
alexbii.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
alexbii	|- ( A. x ph -> ( E. x ps <-> E. x ch ) )

Proof of Theorem alexbii

Step	Hyp	Ref	Expression
1		alexbii.1	. . . 4 |- ( ph -> ( ps <-> ch ) )
2	1	biimpd 219	. . 3 |- ( ph -> ( ps -> ch ) )
3	2	aleximi 1759	. 2 |- ( A. x ph -> ( E. x ps -> E. x ch ) )
4	1	biimprd 238	. . 3 |- ( ph -> ( ch -> ps ) )
5	4	aleximi 1759	. 2 |- ( A. x ph -> ( E. x ch -> E. x ps ) )
6	3, 5	impbid 202	1 |- ( A. x ph -> ( E. x ps <-> E. x ch ) )

Syntax hints:   -> wi 4   <-> wb 196   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:  exbi  1773  exbidh  1794  exintrbi  1818  eleq2d  2687  bnj956  30847