Theorem and4as 10427

Description: /\ associativity.

Assertion

Ref	Expression
and4as	|- ((ph /\ ps /\ (ch /\ th)) <-> ((ph /\ ps /\ ch) /\ th))

Proof of Theorem and4as

Step	Hyp	Ref	Expression
1		pm3.2 283	. . . . 5 |- ((ph /\ ps /\ ch) -> (th -> ((ph /\ ps /\ ch) /\ th)))
2	1	3exp 834	. . . 4 |- (ph -> (ps -> (ch -> (th -> ((ph /\ ps /\ ch) /\ th)))))
3	2	imp4a 364	. . 3 |- (ph -> (ps -> ((ch /\ th) -> ((ph /\ ps /\ ch) /\ th))))
4	3	3imp 829	. 2 |- ((ph /\ ps /\ (ch /\ th)) -> ((ph /\ ps /\ ch) /\ th))
5		id 59	. . . . 5 |- ((ph /\ ps /\ (ch /\ th)) -> (ph /\ ps /\ (ch /\ th)))
6	5	3exp 834	. . . 4 |- (ph -> (ps -> ((ch /\ th) -> (ph /\ ps /\ (ch /\ th)))))
7	6	exp4a 380	. . 3 |- (ph -> (ps -> (ch -> (th -> (ph /\ ps /\ (ch /\ th))))))
8	7	3imp1 848	. 2 |- (((ph /\ ps /\ ch) /\ th) -> (ph /\ ps /\ (ch /\ th)))
9	4, 8	impbi 157	1 |- ((ph /\ ps /\ (ch /\ th)) <-> ((ph /\ ps /\ ch) /\ th))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777

This theorem is referenced by:  elo 10439

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 779

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