Theorem 3anim123i 1187

Description: Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)

Hypotheses

Ref	Expression
3anim123i.1	|- ( ph -> ps )
3anim123i.2	|- ( ch -> th )
3anim123i.3	|- ( ta -> et )

Assertion

Ref	Expression
3anim123i	|- ( ( ph /\ ch /\ ta ) -> ( ps /\ th /\ et ) )

Proof of Theorem 3anim123i

Step	Hyp	Ref	Expression
1		3anim123i.1	. . 3 |- ( ph -> ps )
2	1	3ad2ant1 1021	. 2 |- ( ( ph /\ ch /\ ta ) -> ps )
3		3anim123i.2	. . 3 |- ( ch -> th )
4	3	3ad2ant2 1022	. 2 |- ( ( ph /\ ch /\ ta ) -> th )
5		3anim123i.3	. . 3 |- ( ta -> et )
6	5	3ad2ant3 1023	. 2 |- ( ( ph /\ ch /\ ta ) -> et )
7	2, 4, 6	3jca 1180	1 |- ( ( ph /\ ch /\ ta ) -> ( ps /\ th /\ et ) )

Syntax hints:   -> wi 4   /\ w3a 981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 983

This theorem is referenced by:  3anim1i  1188  3anim2i  1189  3anim3i  1190  syl3an  1292  syl3anl  1301  spc3egv  2865  spc3gv  2866  eloprabga  6034  le2tri3i  8183  fzmmmeqm  10182  elfz1b  10214  elfz0fzfz0  10250  elfzmlbp  10256  elfzo1  10316  flltdivnn0lt  10449  pfxeq  11150  modmulconst  12167  nndvdslegcd  12319  lgsmulsqcoprm  15556