Theorem pm5.21 695

Description: Two propositions are equivalent if they are both false. Theorem *5.21 of [WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)

Assertion

Ref	Expression
pm5.21	|- ( ( -. ph /\ -. ps ) -> ( ph <-> ps ) )

Proof of Theorem pm5.21

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 |- ( ( -. ph /\ -. ps ) -> -. ph )
2	1	pm2.21d 619	. 2 |- ( ( -. ph /\ -. ps ) -> ( ph -> ps ) )
3		simpr 110	. . 3 |- ( ( -. ph /\ -. ps ) -> -. ps )
4	3	pm2.21d 619	. 2 |- ( ( -. ph /\ -. ps ) -> ( ps -> ph ) )
5	2, 4	impbid 129	1 |- ( ( -. ph /\ -. ps ) -> ( ph <-> ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm5.21im  696