Theorem and4com 10433

Description: /\ associativity.

Assertion

Ref	Expression
and4com	|- ((ph /\ (ps /\ ch /\ th)) <-> ((ph /\ ps /\ ch) /\ th))

Proof of Theorem and4com

Step	Hyp	Ref	Expression
1		pm3.2 283	. . . 4 |- ((ph /\ ps /\ ch) -> (th -> ((ph /\ ps /\ ch) /\ th)))
2	1	3exp 832	. . 3 |- (ph -> (ps -> (ch -> (th -> ((ph /\ ps /\ ch) /\ th)))))
3	2	3imp2 848	. 2 |- ((ph /\ (ps /\ ch /\ th)) -> ((ph /\ ps /\ ch) /\ th))
4		pm3.2 283	. . . 4 |- (ph -> ((ps /\ ch /\ th) -> (ph /\ (ps /\ ch /\ th))))
5	4	3expd 850	. . 3 |- (ph -> (ps -> (ch -> (th -> (ph /\ (ps /\ ch /\ th))))))
6	5	3imp1 846	. 2 |- (((ph /\ ps /\ ch) /\ th) -> (ph /\ (ps /\ ch /\ th)))
7	3, 6	impbi 157	1 |- ((ph /\ (ps /\ ch /\ th)) <-> ((ph /\ ps /\ ch) /\ th))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775

This theorem is referenced by:  eeeeanv 10436

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-an 225  df-3an 777

Copyright terms: Public domain