Theorem rnlem 579

Description: Lemma used in construction of real numbers.

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		anandir 393	. 2 |- (((ph /\ ps) /\ (ch /\ th)) <-> ((ph /\ (ch /\ th)) /\ (ps /\ (ch /\ th))))
2		anandi 392	. . 3 |- ((ph /\ (ch /\ th)) <-> ((ph /\ ch) /\ (ph /\ th)))
3		anandi 392	. . 3 |- ((ps /\ (ch /\ th)) <-> ((ps /\ ch) /\ (ps /\ th)))
4	2, 3	anbi12i 369	. 2 |- (((ph /\ (ch /\ th)) /\ (ps /\ (ch /\ th))) <-> (((ph /\ ch) /\ (ph /\ th)) /\ ((ps /\ ch) /\ (ps /\ th))))
5		ancom 333	. . . 4 |- (((ps /\ ch) /\ (ps /\ th)) <-> ((ps /\ th) /\ (ps /\ ch)))
6	5	anbi2i 367	. . 3 |- ((((ph /\ ch) /\ (ph /\ th)) /\ ((ps /\ ch) /\ (ps /\ th))) <-> (((ph /\ ch) /\ (ph /\ th)) /\ ((ps /\ th) /\ (ps /\ ch))))
7		an4 388	. . 3 |- ((((ph /\ ch) /\ (ph /\ th)) /\ ((ps /\ th) /\ (ps /\ ch))) <-> (((ph /\ ch) /\ (ps /\ th)) /\ ((ph /\ th) /\ (ps /\ ch))))
8	6, 7	bitr 151	. 2 |- ((((ph /\ ch) /\ (ph /\ th)) /\ ((ps /\ ch) /\ (ps /\ th))) <-> (((ph /\ ch) /\ (ps /\ th)) /\ ((ph /\ th) /\ (ps /\ ch))))
9	1, 4, 8	3bitr 155	1 |- (((ph /\ ps) /\ (ch /\ th)) <-> (((ph /\ ch) /\ (ps /\ th)) /\ ((ph /\ th) /\ (ps /\ ch))))

Syntax hints:   <-> wb 127   /\ wa 196

This theorem is referenced by:  mulcmpblnr 3977

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

This theorem depends on definitions:  df-bi 128  df-an 198

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