Theorem rnlem 931

Description: Lemma used in construction of real numbers. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
rnlem	|- (((ph /\ ps) /\ (ch /\ th)) <-> (((ph /\ ch) /\ (ps /\ th)) /\ ((ph /\ th) /\ (ps /\ ch))))

Proof of Theorem rnlem

Step	Hyp	Ref	Expression
1		an4 797	. . . 4 |- (((ph /\ ps) /\ (ch /\ th)) <-> ((ph /\ ch) /\ (ps /\ th)))
2	1	biimpi 186	. . 3 |- (((ph /\ ps) /\ (ch /\ th)) -> ((ph /\ ch) /\ (ps /\ th)))
3		an42 798	. . . 4 |- (((ph /\ th) /\ (ps /\ ch)) <-> ((ph /\ ps) /\ (ch /\ th)))
4	3	biimpri 197	. . 3 |- (((ph /\ ps) /\ (ch /\ th)) -> ((ph /\ th) /\ (ps /\ ch)))
5	2, 4	jca 518	. 2 |- (((ph /\ ps) /\ (ch /\ th)) -> (((ph /\ ch) /\ (ps /\ th)) /\ ((ph /\ th) /\ (ps /\ ch))))
6	3	biimpi 186	. . 3 |- (((ph /\ th) /\ (ps /\ ch)) -> ((ph /\ ps) /\ (ch /\ th)))
7	6	adantl 452	. 2 |- ((((ph /\ ch) /\ (ps /\ th)) /\ ((ph /\ th) /\ (ps /\ ch))) -> ((ph /\ ps) /\ (ch /\ th)))
8	5, 7	impbii 180	1 |- (((ph /\ ps) /\ (ch /\ th)) <-> (((ph /\ ch) /\ (ps /\ th)) /\ ((ph /\ th) /\ (ps /\ ch))))

Syntax hints:   <-> wb 176   /\ wa 358

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-an 360

This theorem is referenced by: (None)