Theorem pm5.55 674

Description: Theorem *5.55 of [WhiteheadRussell] p. 125.

Assertion

Ref	Expression
pm5.55	|- (((ph \/ ps) <-> ph) \/ ((ph \/ ps) <-> ps))

Proof of Theorem pm5.55

Step	Hyp	Ref	Expression
1		pm5.13 663	. 2 |- ((ps -> ph) \/ (ph -> ps))
2		pm4.72 640	. . . 4 |- ((ps -> ph) <-> (ph <-> (ps \/ ph)))
3		orcom 246	. . . . 5 |- ((ps \/ ph) <-> (ph \/ ps))
4	3	bibi2i 607	. . . 4 |- ((ph <-> (ps \/ ph)) <-> (ph <-> (ph \/ ps)))
5		bicom 519	. . . 4 |- ((ph <-> (ph \/ ps)) <-> ((ph \/ ps) <-> ph))
6	2, 4, 5	3bitr 177	. . 3 |- ((ps -> ph) <-> ((ph \/ ps) <-> ph))
7		pm4.72 640	. . . 4 |- ((ph -> ps) <-> (ps <-> (ph \/ ps)))
8		bicom 519	. . . 4 |- ((ps <-> (ph \/ ps)) <-> ((ph \/ ps) <-> ps))
9	7, 8	bitr 173	. . 3 |- ((ph -> ps) <-> ((ph \/ ps) <-> ps))
10	6, 9	orbi12i 257	. 2 |- (((ps -> ph) \/ (ph -> ps)) <-> (((ph \/ ps) <-> ph) \/ ((ph \/ ps) <-> ps)))
11	1, 10	mpbi 189	1 |- (((ph \/ ps) <-> ph) \/ ((ph \/ ps) <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225

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