Theorem impel 280

Description: An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.)

Hypotheses

Ref	Expression
impel.1	|- ( ph -> ( ps -> ch ) )
impel.2	|- ( th -> ps )

Assertion

Ref	Expression
impel	|- ( ( ph /\ th ) -> ch )

Proof of Theorem impel

Step	Hyp	Ref	Expression
1		impel.2	. . 3 |- ( th -> ps )
2		impel.1	. . 3 |- ( ph -> ( ps -> ch ) )
3	1, 2	syl5 32	. 2 |- ( ph -> ( th -> ch ) )
4	3	imp 124	1 |- ( ( ph /\ th ) -> ch )

Syntax hints:   -> wi 4   /\ wa 104

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

This theorem is referenced by:  pm4.55dc  946  fiintim  7123  eqinfti  7219  finomni  7339  frecuzrdgrclt  10678  seq3coll  11107  swrdswrd  11290  swrdccatin1  11310  swrdccatin2  11314  fprodsplitsn  12212  nninfctlemfo  12629  unct  13081  isnzr2  14217  dvcnp2cntop  15442  fsumdvdsmul  15734  perfectlem2  15743  upgrwlkcompim  16232  wlkv0  16239  trlsegvdeglem1  16330