Theorem pm4.72 769

Description: Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jan-2013.)

Assertion

Ref	Expression
pm4.72	|- ( ( ph -> ps ) <-> ( ps <-> ( ph \/ ps ) ) )

Proof of Theorem pm4.72

Step	Hyp	Ref	Expression
1		olc 664	. . 3 |- ( ps -> ( ph \/ ps ) )
2		pm2.621 698	. . 3 |- ( ( ph -> ps ) -> ( ( ph \/ ps ) -> ps ) )
3	1, 2	impbid2 141	. 2 |- ( ( ph -> ps ) -> ( ps <-> ( ph \/ ps ) ) )
4		orc 665	. . 3 |- ( ph -> ( ph \/ ps ) )
5		bi2 128	. . 3 |- ( ( ps <-> ( ph \/ ps ) ) -> ( ( ph \/ ps ) -> ps ) )
6	4, 5	syl5 32	. 2 |- ( ( ps <-> ( ph \/ ps ) ) -> ( ph -> ps ) )
7	3, 6	impbii 124	1 |- ( ( ph -> ps ) <-> ( ps <-> ( ph \/ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 103   \/ wo 661

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  bigolden  896  ssequn1  3142