Theorem rnlem 917

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 550	. . . 4 |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ( ph /\ ch ) /\ ( ps /\ th ) ) )
2	1	biimpi 118	. . 3 |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ( ( ph /\ ch ) /\ ( ps /\ th ) ) )
3		an42 551	. . . 4 |- ( ( ( ph /\ th ) /\ ( ps /\ ch ) ) <-> ( ( ph /\ ps ) /\ ( ch /\ th ) ) )
4	3	biimpri 131	. . 3 |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ( ( ph /\ th ) /\ ( ps /\ ch ) ) )
5	2, 4	jca 300	. 2 |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ( ( ( ph /\ ch ) /\ ( ps /\ th ) ) /\ ( ( ph /\ th ) /\ ( ps /\ ch ) ) ) )
6	3	biimpi 118	. . 3 |- ( ( ( ph /\ th ) /\ ( ps /\ ch ) ) -> ( ( ph /\ ps ) /\ ( ch /\ th ) ) )
7	6	adantl 271	. 2 |- ( ( ( ( ph /\ ch ) /\ ( ps /\ th ) ) /\ ( ( ph /\ th ) /\ ( ps /\ ch ) ) ) -> ( ( ph /\ ps ) /\ ( ch /\ th ) ) )
8	5, 7	impbii 124	1 |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ( ( ph /\ ch ) /\ ( ps /\ th ) ) /\ ( ( ph /\ th ) /\ ( ps /\ ch ) ) ) )

Syntax hints:   /\ wa 102   <-> wb 103

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

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)