Theorem 3jcad 1173

Description: Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)

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 123	. . 3 |- ( ( ph /\ ps ) -> ch )
3		3jcad.2	. . . 4 |- ( ph -> ( ps -> th ) )
4	3	imp 123	. . 3 |- ( ( ph /\ ps ) -> th )
5		3jcad.3	. . . 4 |- ( ph -> ( ps -> ta ) )
6	5	imp 123	. . 3 |- ( ( ph /\ ps ) -> ta )
7	2, 4, 6	3jca 1172	. 2 |- ( ( ph /\ ps ) -> ( ch /\ th /\ ta ) )
8	7	ex 114	1 |- ( ph -> ( ps -> ( ch /\ th /\ ta ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 975

This theorem is referenced by:  ixxssixx  9859  iccid  9882  fzen  9999