Theorem pm4.72 846

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 373	. . 3 |- (ps -> (ph \/ ps))
2		pm2.621 397	. . 3 |- ((ph -> ps) -> ((ph \/ ps) -> ps))
3	1, 2	impbid2 195	. 2 |- ((ph -> ps) -> (ps <-> (ph \/ ps)))
4		orc 374	. . 3 |- (ph -> (ph \/ ps))
5		bi2 189	. . 3 |- ((ps <-> (ph \/ ps)) -> ((ph \/ ps) -> ps))
6	4, 5	syl5 28	. 2 |- ((ps <-> (ph \/ ps)) -> (ph -> ps))
7	3, 6	impbii 180	1 |- ((ph -> ps) <-> (ps <-> (ph \/ ps)))

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-or 359

This theorem is referenced by:  bigolden  901  ssequn1  3434  ssunsn2  3866