Theorem 3jcad 818

Description: Deduction conjoining the consequents of three implications.

Hypotheses

Ref	Expression
3jcad.1	|- (ph -> (ps -> ch))
3jcad.2	|- (ph -> (ps -> th))
3jcad.3	|- (ph -> (ps -> ta))

Assertion

Ref	Expression
3jcad	|- (ph -> (ps -> (ch /\ th /\ ta)))

Proof of Theorem 3jcad

Step	Hyp	Ref	Expression
1		3jcad.1	. . . 4 |- (ph -> (ps -> ch))
2	1	imp 350	. . 3 |- ((ph /\ ps) -> ch)
3		3jcad.2	. . . 4 |- (ph -> (ps -> th))
4	3	imp 350	. . 3 |- ((ph /\ ps) -> th)
5		3jcad.3	. . . 4 |- (ph -> (ps -> ta))
6	5	imp 350	. . 3 |- ((ph /\ ps) -> ta)
7	2, 4, 6	3jca 817	. 2 |- ((ph /\ ps) -> (ch /\ th /\ ta))
8	7	ex 373	1 |- (ph -> (ps -> (ch /\ th /\ ta)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 773

This theorem is referenced by:  climmulc2 7065  clim2serzt 7070  iscau3 7876  caussi 7889  lmcau 7930  cnlnssadj 9928

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 775

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