Theorem pm4.72

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)

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

Proof

Step	Hyp	Ref	Expression
1		olc	|- ( ps -> ( ph \/ ps ) )
2		pm2.621	|- ( ( ph -> ps ) -> ( ( ph \/ ps ) -> ps ) )
3	1 2	impbid2	|- ( ( ph -> ps ) -> ( ps <-> ( ph \/ ps ) ) )
4		orc	|- ( ph -> ( ph \/ ps ) )
5		biimpr	|- ( ( ps <-> ( ph \/ ps ) ) -> ( ( ph \/ ps ) -> ps ) )
6	4 5	syl5	|- ( ( ps <-> ( ph \/ ps ) ) -> ( ph -> ps ) )
7	3 6	impbii	|- ( ( ph -> ps ) <-> ( ps <-> ( ph \/ ps ) ) )